Блог Анкара

In This Chapter

^ Checking out your membership card and other important info from your plan

^ Determining when your coverage begins

^ Getting organized — why keeping records pays off

■ XOu’re in. Medicare has confirmed your enrollment, and you’re going to Jt Get drug coverage under Part D. Whether that makes you feel elated, relieved, or wary, you’ve overcome the hurdle of making several important, and perhaps difficult, decisions. Congratulate yourself!

So what happens next? This chapter explains what to expect from your drug plan the first time out. I begin with the question of when your drug coverage actually starts, according to your circumstances. Then I describe what you’ll receive from your plan — namely, your membership card and a lot of important documents to read. I also explain what to do if you’re told you need to pay a late penalty. Finally, I clue you in on why keeping good records pays off in tracking your coverage and expenses, and in protecting yourself in case of disputes. Even if this isn’t your first time in a Medicare prescription drug plan, you may find some of this information helpful.

Knowing When Your Coverage Will Start

Can’t wait to get that drug coverage? It’s what you’ve signed up for, after all. You won’t have to wait long — less than one month in most cases. But your coverage doesn’t start until the date it becomes Effective, Meaning the very first day you can fill a prescription under your Part D plan, even if you receive your membership card in the mail before then.

When you’re joining a Part D plan for the first time, the date your coverage begins depends on the following circumstances:

You’re just coming into the Medicare program because you’re turning 65 or qualifying through disability.

You’re just coming into the Medicare program after a delay due to turning 65 while living abroad or in prison

You’ve received a special enrollment period because you lost your creditable drug coverage from an employer or some other plan, or because you recently came out of prison or have returned to the U. S. after a period living abroad.

You failed to join Part D when you were first eligible and now need to enroll in a drug plan during the annual enrollment period at the end of the year.

You qualify for Part D’s Extra Help because your income is limited. Table 13-1 shows when your coverage begins in all of these situations.

If you sign up for a plan during your initial Medicare enrollment period upon turning 65

On the first day of the month in which you turn 65, if you enrolled in the plan during the previous three months. Or on the first day of the month after enrollment if you signed up in the month of your birthday, or during the following three months.

If you sign up for a plan because of disability during your initial Medicare enrollment period

On the first day of the month in which you receive your 25th disability payment, if you enrolled in the plan during the previous three months. Or on the first day of the month after enrollment if you signed up in the month you received your 25th check, or during the following three months.

If you sign up for a plan during a special initial enrollment period due to turning 65 while abroad or in prison

Immediately after your return to the U. S. or your release from prison, if you enrolled in the plan during the previous three months. Or on the first day of the month after enrollment if you signed up in the month of your return or release, or during the following three months.

As you can see in Table 13-1, the longest you can wait to start coverage after enrollment is six weeks. And that’s only in one particular situation: if you delay joining Part D beyond when you were first eligible and need to wait for open enrollment at the end of the year. If you sign up on the first day of that period, November 15, your coverage starts six weeks later on January 1.

The shortest wait can be as little as one day. If you enroll in a plan on the last day of a month, your coverage becomes effective the first day of the following month — in other words, the next day. (However, I don’t advise cutting it that fine. Loading your coverage info into Medicare’s computer system takes a while, so enrolling at the last minute can cause delays when filling your first prescriptions, as explained in Chapter 14.)

Of course, if you’re already in Part D, your current coverage lasts until the end of the calendar year — midnight on December 31. If you decide to switch plans and sign up for another during open enrollment (November 15 through December 31), coverage in your new plan starts January 1. (See Chapter 17 for details about switching plans.) If you stay in the same plan, your coverage just rolls on through the following year.

Receiving Your Plan’s Card and Other Important Stuff You Need to Read

Soon after you enroll, your plan will send you a bunch of stuff through the mail. So watch for its arrival and get out your glasses! This is important information you must read — even if you’ve been in a Medicare drug or private health plan before. In the next several sections, I show you what to expect and check up on when you receive your plan’s membership card, its Evidence of Coverage document, and other information. I also touch on the form you should send back to the plan showing what other types of coverage you may have for prescription drugs or medical care. And I explain what to do if you’re told you need to pay a late penalty on your Medicare prescription drug coverage.

JfitBER You receive a membership card and information about coverage regardless ^ of whether you’re joining Part D for the first time, have just switched from

Jjljjl J one drug plan to another, or are remaining in the same plan you had last year. In this last situation, even though you don’t need to re-enroll, you still get a new card and Evidence of Coverage (EOC) document for the new year. That’s because some details of your coverage have likely changed since the previous year. (Your plan sends advance warning of any changes in its Annual Notice of Change [ANOC], mailed in the fall, and incorporates them in your new EOC document. So if you didn’t read the ANOC, be sure to read the EOC.)

The membership ID card: Your key to coverage

The very first mailing you receive from your new Part D plan is an acknowledgment of your enrollment request, which the plan must send you within ten calendar days of receiving it. This mailing also tells you when you can expect your coverage to become effective, as explained in the previous section. The plan sends your membership ID card either with this mailing or soon afterward, and you can start using it at the pharmacy as soon as your coverage begins.

BER If you don’t receive your card by the time your coverage starts, you can use what your plan calls its Proof of coverage To fill your prescriptions until your card arrives. This proof may be a copy of your enrollment form, the plan’s acknowledgment of your enrollment request, or a letter from the plan stating that you’re entitled to coverage, starting on the effective date. (See Chapter 14 on what to do if you encounter problems obtaining your prescriptions the first time out.) Note that Proof of coverage Isn’t the same as the Evidence of Coverage document, which may arrive either with your membership card or several weeks later.

When you receive your plan’s membership card, keep it safe. The card is your key to getting prescription drugs and letting your pharmacist know what you should pay, as I explain in Chapter 14. You need to present it at the pharmacy each time you fill a prescription. So keep it in your wallet. (If you were in a different Part D plan previously, destroy your old card — but be sure to do so only after your new coverage begins.)

In the following sections, I explain the details you need to check when you receive your card and give you pointers on how to keep track of different Medicare cards you may have.

Double-checking the details

HHJ ) Verify that the information on your card is correct. It should include W The plan’s name

W Your name and membership ID number W The plan’s customer service phone number(s) W The plan’s mailing address

Make sure this plan is the one you signed up for. If you’re not certain, take a look at your plan’s info packet (which it must send you soon after it receives your enrollment request, ideally before your coverage begins). You can also check the plan’s name and identification number that appears on your membership card against information in your Medicare & You Handbook or on Medicare’s online plan finder tool. Or you can call Medicare’s help line to check what kind of plan it is. If it turns out you’re enrolled in the wrong plan — for example, this plan’s a private Medicare health plan rather than the drugs-only plan you wanted — flip to Chapter 17 to find out what to do.

Playing your cards right

You may find yourself with several cards entitling you to different Medicare services. And sometimes it isn’t easy to see at a glance which card is which, or to remember what each is used for. It can be especially confusing if, for example, you have a prescription plan and a Medigap policy that are provided by the same insurance company, and its name appears on both cards.

Presenting the correct card when you show up at a doctor’s office or hospital, or use any other medical service, is critically important. That’s because the card tells your provider whom to bill. For example, if you’re in a Medicare private health plan, you must show the plan’s card and not your Medicare ID card. If by mistake you show your Medicare card instead, the provider will bill Medicare and not the plan. Medicare will then deny your claim, and you or the provider will have a lot of hassle sorting it all out. So it pays to know your cards and use them appropriately.

Here’s how to identify and use each of the Medicare cards you may have:

W Your red-white-and-blue Medicare ID card: Use this card to obtain medical services if you receive your health benefits through traditional Medicare. The card shows your name and Medicare ID number. It says whether you’re entitled to Medicare Part A, (hospital services) or Part B (doctor visits and other outpatient care), or both, as well as the date(s) on which your coverage became effective. The card also provides the phone number of the Medicare help line.

W Your Medicare private health plan card: Use this card (not Your Medicare ID card) to get medical services if you’re enrolled in a Medicare private health plan (such as an HMO, PPO, Special Needs Plan, Private Fee-for-Service plan, a Medicare Cost plan, or a Medicare Medical Savings Account; see Chapter 9 for details on these plans). If your plan includes drug coverage, you may be able to use the same card at the pharmacy when filling your prescriptions, or you may be given a separate card to use at the pharmacy, depending on the plan. Either way, the plan’s name, its identification number, and contact information appears on the card. So does your name and membership number.

W Your Medicare stand-alone prescription drug plan card: Use this card to get your prescriptions filled under Part D if you receive your medical benefits from traditional Medicare or a Medicare private health plan that doesn’t cover drugs (Medicare Medical Savings Accounts, as well as some Private Fee-for-Service plans and Medicare Cost plans). The card shows your name, your plan membership number, the plan’s name, its identification number, and its contact information. Your card likely has wording that indicates you’re entitled to Medicare prescription drug coverage. It may simply say PDP — initials that stand for Prescription Drug Plan, the phrase Medicare uses for stand-alone Part D plans.

W Your Medigap supplementary insurance card: Use this card to prove that you have separate insurance to help cover your co-pays for medical services when you’re enrolled in traditional Medicare. Show it every time you receive services from a doctor, hospital, or other provider. (Medigap insurance can’t be used to cover out-of-pocket costs in Medicare private health plans, as explained in Chapter 9. Nor can it be used for prescription drug expenses, as explained in Chapter 15.) The card shows your name and membership number, the name of the insurance company, its contact information, and the type of Medigap plan you have (marked with a letter of the alphabet, A Through L) — for example, MEDIGAP F.

Information about your plan: Your new bedtime reading material

Sooner or later (either together with your plan’s membership ID card or separately) you’ll receive a sizeable information packet that should include the following documents about your new plan:

W Evidence of Coverage: This booklet is your legal contract with the plan, so be sure to keep it in a safe place. It contains masses of stuff you can use for reference when you first join the plan and throughout the year. Here are some examples:

• Phone numbers: Call if you have any questions or problems.

• An explanation of how the plan works, its conditions, and its rules: This part lays out the plan’s responsibilities in giving you coverage and the rules you must accept. It shows the plan’s costs and benefits for this year, which may be different from those the same plan provided last year.

• Details of the plan’s benefits and costs: If this is a drugs-only plan, the Evidence of Coverage explains what the plan charges for different kinds of covered drugs. If it’s a Medicare private health plan, you’ll find this info, plus details of how much you’ll pay for every covered medical service, like doctor visits, hospital stays, and many more.

• An explanation of your legal rights if you have a complaint against the plan or disagree with a decision it makes: This section includes detailed instructions on how to file a complaint or make an appeal. (I cover the general process of doing both in

Chapter 19.)

W The plan’s drug formulary: The Formulary Is a list of all the medications the plan covers. It shows which drugs come with restrictions such as prior authorization, quantity limits, and step therapy. (I explain what these restrictions mean, and what you can do about them, in Chapter 4.)

W The plan’s pharmacy network: The Pharmacy network Is a list of all the pharmacies in your area that accept your plan’s card. This list shows which ones are Preferred Pharmacies (where your drugs may cost less) or Specialist Pharmacies (which stock special drugs, like those that are injected or require careful handling), as explained in Chapter 14.

W The plan’s provider directory: If your plan is a Medicare managed care plan (HMO, PPO, Special Needs Plan, or Medicare Cost plan), this Provider directory Is a list of doctors, hospitals, and other facilities in your area that are in the plan’s network and have agreed to treat its members.

W The plan’s service area: The service area is a list of all the zip codes that your plan (if it’s a Medicare managed care plan) covers.

Everything in your plan’s info packet is critical. If any of the items listed here are missing, call the plan and ask for them, according to your type of plan:

W If you’re in a Medicare managed care plan that covers prescription drugs, you need all these items. (Private Fee-for-Service plans and Medicare Medical Savings Accounts don’t have limited service areas or provider directories.)

W If you’re in a stand-alone plan that covers only prescription drugs, you just need the first three items on the preceding list.

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A form for disclosing other coverage: Your chance to put it all out there

Your plan’s packet includes a form asking whether you have other coverage for prescription drugs and/or medical care. The plan may have already requested this information on your enrollment form, as explained in Chapter 12, but the separate form in your info packet probably requires more details. Filling out this form and returning it to the plan as instructed are in your best interests.

The plan needs to know of other coverage so all of your benefits can be coordinated properly. This Coordination of benefits Means you don’t pay more than you should, and, in most cases, you don’t have to make separate claims, as explained in Chapter 14. So if you’re entitled to drug and/or medical coverage from any of the following, enter that on the form:

W A current or former employer or union (or COBRA insurance)

W The Veterans Affairs (VA or CHAMPVA) health program

W The Department of Defense (TRICARE)

W The Federal Employees Health Benefits program (FEHB)

W The Indian Health Service, a Tribal Health organization, or the Urban Indian Health Program

W A qualified State Pharmacy Assistance Program (SPAP)

W An individual health insurance policy

I explain each of these programs, and how they fit in with Medicare prescription drug coverage, in Chapter 6.

A late penalty assessment: Your price for missing your enrollment deadline

Not long after you enroll in Part D for the first time, you may receive a letter from your plan saying you need to pay a late penalty. This scenario should happen Only If you miss your deadline for joining Part D and go for more than 63 days without Creditable coverage — drug coverage that’s at least as good as Part D, such as you may have had from an employer or union. (If you haven’t the faintest idea what I’m talking about here, you need to read Chapter 6 to get familiar with creditable coverage and Chapter 8 to find out about the late penalty — and fast!)

How your plan decides whether you should be penalized

Every plan is responsible for finding out whether any new enrollees should have a late penalty. Plans can do so by obtaining the information from Medicare, making their own inquiries, or sending an Attestation form To new enrollees. If you receive this form, you have up to 30 days to respond and indicate whether you’ve had creditable coverage for drugs and, if so, where you got it from.

This is the time when any notices you’ve kept about former creditable coverage come in very handy, as covered in Chapter 6. However, if you’ve lost this proof, be aware that Medicare requires your plan to accept letters from any former employer or union confirming the creditable coverage you once had. So when you return your attestation form, be sure to add the name of and contact info for your former employer or union and, if possible, the dates when your creditable coverage began and ended so your plan can verify them. Or, if you prefer, you can ask the benefits department of your former employer or union to send you a similar letter, and you can attach a copy to your attestation form. (But be careful not to miss that important 30-day deadline for returning the form. Send it in on time, even if the requested letter hasn’t arrived.)

If the plan decides you have an unexplained gap in your drug coverage, it informs Medicare, and someone there does the math to work out the amount of your late penalty. The plan then notifies you of this amount and tells you how it was calculated (according to a formula I explain in Chapter 8).

How your plan may be wrong

What if your plan slaps you with a late penalty you don’t think you deserve? That’s a real curve ball. But your plan may have gotten it wrong. Here are some of the ways mistakes can happen:

W Your record of coverage hasn’t been verified due to computer glitches or other screw-ups in the system. People have occasionally received these letters after being in Part D ever since it began — and therefore couldn’t possibly deserve a late penalty! Gotta love technology.

W You had drug coverage but didn’t know it wasn’t creditable because the plan that provided it didn’t clearly inform you of this fact.

W During the time in question, you were living abroad or in prison and therefore couldn’t sign up for Part D. Provided you started receiving Part D coverage within one of the special enrollment periods granted after your return or release, as explained in Chapter 12, you shouldn’t face a late penalty.

W You went without creditable coverage for a certain length of time, but not for as many months as the letter claims. The length of time without coverage determines the amount of your late penalty, as explained in

Chapter 8.

W You qualified for Extra Help (see Chapter 5) and joined a Part D plan

Between May 15, 2006, and December 31, 2008. The late penalty was waived during this period for people eligible for Extra Help and without creditable coverage. Congress has now made this waiver permanent, so starting on January 1, 2009, nobody eligible for Extra Help faces a late penalty.

W During the time in question, you were living in an area affected by Hurricane Katrina. Medicare allowed a special enrollment period extending from May 15 to December 31, 2006, and waived the late penalty for this six-month period for people living in certain parishes on the Gulf Coast when the hurricane hit in August 2005.

What your rights are if you think your plan is Wrong

When your plan sends you a letter regarding your late penalty amount, it encloses a notice headed "Your Right to Ask Medicare to Review Your Part D Late Enrollment Penalty," which explains your right to challenge the ruling. It also encloses a form you can use to request a Reconsideration — in other words, an independent review of the decision. You have 60 days from the date on the letter to complete and return the form to the address provided. (You can authorize someone else to fill out the form on your behalf if you want; follow the instructions on the form.) If you choose to request a reconsideration, you must still pay the late penalty throughout the process until a decision is made.

On the form, check off any circumstances that apply to you or write other reasons on a separate sheet. Add copies (not The originals) of any documents that support your case, which may include proof of creditable coverage from a former employer or union, or a previous plan’s benefits summary that didn’t explain whether the plan’s drug coverage was creditable, for example.

Be sure to meet all deadlines or, if you have a good reason why you can’t, request an extension by following the instructions on the form. Otherwise, your case will be dismissed, and you’ll have no further opportunity to argue it.

You should receive a decision from the Independent Review Entity (IRE), the official panel that conducts the reconsideration, within 90 days of the IRE receiving your request. (The decision may come a lot sooner, but it depends on how many similar cases the IRE is dealing with.) The IRE can extend the reconsideration process for up to 14 days for good reason, such as to examine more evidence. Here’s what happens next, based on the IRE’s decision:

W If the decision goes in your favor: You no longer face a late penalty. Your plan must then refund any late penalty fees you’ve already paid during the reconsideration process. Or, if the penalty is reduced, the plan must refund any overpayments you’ve made.

W If the decision goes against you: You’re stuck with the late penalty amount. You can’t appeal against a negative decision.

How to pay the late penalty

If your plan premiums are deducted from your monthly Social Security check, then the late penalty is automatically taken out of it, too. If the plan sends you a monthly bill for premiums, the late penalty is added to the bill. If you prefer, you can choose to pay the penalty on a quarterly or annual basis.

Nice ‘n’ Neat: Starting and Keeping Careful Records

People tend to be pretty slapdash about keeping records. Overwhelmed by junk mail, sometimes you toss the lot and lose something important. You put a vital document in a folder and then forget where you put the folder. Or you start off with the best intentions to sort and keep records in a file cabinet or computer file, and then — just like New Year resolutions that evaporate come February — you somehow fail to update them.

But if there’s one set of records you should keep safe, accessible, complete, and up-to-date, it’s all the accumulated bits of paper relating to your medical and prescription drug insurance. Doing so helps you keep track of your expenses and see how your drug coverage is panning out through the year (how close you’re coming to the doughnut hole, for example; see Chapter 15 for more on this topic). It also serves as a protection in case you get into any disputes with your plan and need hard facts to argue your case. In the following sections, I share how to tidy up your medical and drug insurance records, including how to store them online (if you’re so inclined).

Keeping hard-copy records you can rely on

If you hit a pothole while cruising down the Part D highway, you don’t want to have to scramble for your paperwork only to find it’s missing or unreadable because Fido used it as a new chew toy. In the next several sections, I present some suggestions for keeping records in such a way that you can actually find them when you need them.

Filing each type of insurance separately

Depending on how many different kinds of insurance you have — traditional Medicare, a stand-alone Part D plan, a Medigap supplemental insurance policy, a Medicare private health plan that includes drug coverage, or any other insurance (such as retiree or veterans health benefits) — keep records for each one in a separate file. Even if you have only one plan, creating one file for its medical benefits and another for its drug coverage is practical. Label each folder with the name of the plan and the type of insurance.

Keeping your Evidence of Coverage and other plan information

Your Evidence of Coverage booklet is a legal document that contains details of your legal rights and how to exercise them if you need to, as explained in the earlier section, "Information about your plan: Your new bedtime reading material." If you have more than one plan, file each EOC and other documents in the appropriate folder for each plan so you can access them easily.

Tracking your expenses and level of coverage

Your plan must send a regular statement, called the Explanation of Benefits, About the treatment you’ve received and what you’ve paid. In the case of Part D, every EOB is worth keeping because together they show you

W How much you and your plan are paying for your drugs as the months go by. (The total amount paid by you and your plan affects your coverage level.)

W How close you are to the doughnut hole and, if your costs are high enough to land you in it, how much more you need to spend out of pocket before you qualify for the low costs of catastrophic coverage. (I explain the doughnut hole, catastrophic coverage, and other coverage levels in Chapter 15.)

W How much you’ve spent out of pocket on drugs during the year to date in this plan — in case you need to switch to another plan partway through the year (as explained in Chapter 17) and want to ensure that all of your payments under the old plan will count toward your out-of-pocket limit in the new one.

Hanging on to documents that may help you resolve problems

Medicare has a system consumers can use to resolve disputes with their plans — whether they involve complaints against traditional Medicare, private Medicare health plans, or Part D plans — at several levels of appeal. (I explain the procedures for filing grievances and appeals in Chapter 19.) In Part D, you may need to ask the plan to cover a nonformulary drug or waive a restriction.

Keeping records of any interaction with your plan — including notes from phone conversations — helps your case if you need to file an appeal.

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Opening and reading your mail

When an envelope marked with your plan’s name shows up in your mail box, open it! This advice may sound obvious, but plans too often send unwanted marketing materials to their members (sometimes unrelated to the health or prescription drug plan they’re on), and it’s easy to get into the habit of disregarding everything. But some plan mailings are important and may require timely action on your part. Here are some examples:

W The plan notifies you that it’s going to stop covering one of the drugs you’re taking and gives you 60 days’ warning. Or it informs you that one of your drugs has been taken off the market for safety reasons, and the plan will no longer cover it.

W The plan alerts you that it’s disenrolling you and terminating your coverage for some reason (as explained in Chapter 17) and gives you a certain amount of time to respond.

W The plan has decided you should pay a late penalty (as explained earlier in this chapter) and gives you 60 days to respond.

W The plan sends you its Annual Notice of Change (ANOC) — as all plans must do in October — which shows how its costs and benefits, and maybe its whole design, will change for the following year. (I explain more in Chapter 17 about the critical importance of reading the ANOC Every year.)

W The plan informs you that it’s withdrawing service from your area, not renewing its contract with Medicare, or going out of business entirely. (See Chapter 17 for more on these possibilities.)

Tracking information online

Everything’s going electronic, and Medicare’s no exception. Following are two ways you may be able to keep track of your medical information online, up to a point, if that’s your preference:

W On Medicare’s Web site: Medicare offers a free way of accessing some of your personal medical information through its MyMedicare Web portal. Among other tasks, you can use this tool to

• Track your health claims in traditional Medicare

• See which preventive tests and screenings you’re entitled to

• Order a replacement for a lost Medicare card

• Keep a list of your medications

To create a personal account, go to Www. MyMedicare. gov, click "Need to Register?," and enter your Medicare ID number. About two weeks later, Medicare will send you the password you need to access your account. Oddly, the password is sent via regular mail!

W In a personal health record: An increasing number of Medicare drug and private health plans are offering personal health records (PHRs) to their members for free. A PHR Is an online tool that allows you to enter and keep any information you choose about

• Your medical history

• The visits you make to doctors and other medical services

• The dates of tests and screenings you need for your health condition

Call your plan or visit its Web site to see whether it offers a PHR. You can also download a PHR that’s not connected to any health plan for free from the Internet.

These personal accounts may be a convenient way of storing files that are of interest to you — information downloaded from the Internet about your health condition, for example — or keeping a record of your treatments and setting up a calendar for your medical appointments. In some cases, you may be able to arrange for e-mails alerting you to tests, screenings, and checkups that are due. As more doctors transfer to electronic recordkeeping, you may be able to arrange to have their records of your visits, diagnoses, and tests transferred to your account. In this way, you choose what information you want to put into your account.

On the flip side, neither the MyMedicare site nor PHRs provide a way of tracking all your medical records and expenses as comprehensively as you can do the old-fashioned way — by filing paper. At least not yet. Also, many people remain uncomfortable with committing sensitive health information to a Web site. Both the operators of the MyMedicare portal and the suppliers of PHRs (including some of those private Medicare health plans) maintain that the accounts are secure and can be accessed only with your permission. But look carefully at the contract agreement (which you must sign) to see whether they retain the right to share your information with other parties.

Chapter 14

In This Chapter

► Creating lists of consecutive integers from descriptions

► Focusing on consecutive odds and evens

► Solving for one of several in a list

► Applying consecutive integers to practical problems

Ou became familiar with your first list of consecutive integers when you were taught to count. You may not remember when you held up fingers and counted, "One, two, three. . . ." But we all did it — except maybe Einstein, who probably counted by threes.

In this chapter, you find consecutive integers, consecutive even integers, consecutive multiples of fives, and so on. The word problems come in as puzzles to find the first, the middle, or the last in a list of consecutive integers. After an introduction on ways to find the sum of a large number of consecutive numbers, you’ll see some interesting applications from seating charts to orchards.

Adding Up Sets of Consecutives

A list of consecutive integers may be one of the following:

3, 4, 5, 6, 7: Five consecutive integers starting with 3 -12, -10, -8, -6: Four consecutive even integers ending with -6 15, 20, 25, 30, 35: Five consecutive multiples of 5 starting with 15 0, 4, 8, 12, 16: Five consecutive multiples of 4, with middle 8

Any list of consecutive integers can be described in many different ways. You may give the overall pattern that describes how far apart the numbers in the list are, and then you give one of the numbers in the list and its position. Or

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You may give the rule on how far apart the numbers are and then some characteristic of the list. You may give the first and last numbers and tell how many are in the list. All you need is enough information to distinguish your list from any others.

Writing the list algebraically

A list of consecutive integers consists of numbers that are all the same distance apart. You structure the distance apart by adding some constant value repeatedly, until you have the desired number of integers in the list. For example, if you want a list of the first six multiples of 3, starting with the number 0, you find the numbers by starting with 0, then adding 3 to 0 to get 3, then adding another 3 to 3 to get 6, and so on. Here’s how to create that list: 0, 0 + 3, 0 + 3 + 3, 0 + 3 + 3 + 3, 0 + 3 + 3 + 3 + 3, 0 + 3 + 3 + 3 + 3 + 3. Simplifying these terms, I write them: 0, 0 + 3, 0 + 6, 0 + 9, 0 + 12, 0 + 15. You’re probably wondering why I can’t take that final leap and add the zeros in. You’ll see my point shortly.

When writing a list of consecutive integers, you let the first integer in the list be signified with a variable. Some people like to use X, Because that’s the universal unknown or variable. In this chapter, you see a consistent use of another variable, N. Using N For consecutive integers is pretty standard notation, too.

So, if the first integer in the list is N, Then the next integer in the list is N + something. The Something Is that common difference between the terms in the list. Writing a list of any six multiples of 3, I start with N, Then N + 3, N + 6, N + 9, N + 12, and finally N + 15. You see that the list hinges on two things:

E That the terms are all the same distance apart That the first integer is what you say it is

I declared, here, that N Is a multiple of 3. I can do that! You Can do that!

Following are more examples of the algebraic representation of lists of consecutive integers. You choose the number for N, And the rest fall in line. Of course, if you want even integers, you have to pick an even integer for N:

E N, n + 2, N + 4, N + 6: Four consecutive even integers

E N, n + 2, N + 4, N + 6: Four consecutive odd integers (Yes, this list can be even or odd. It all depends on what that first N Is.)

E N, n + 5, N + 10, N + 15: Four consecutive multiples of 5

E N, n + 14, N + 28: Three consecutive multiples of 14

E N, n + 1, N + 2, N + 3, N + 4: Five consecutive integers

Reconstructing a list

When solving consecutive integer problems, you solve for N And use that value to answer some question about the list or some number on that list. For example, assume that you’ve solved an equation and found that N = 4. Using that value of N, You can then reconstruct a list of numbers by substituting in the 4 for the N‘s. Or you can answer a question about the third number in the list or some such thing.

If N = 4 and you’re trying to find Three consecutive integers, Then your three integers are: 4, 5, and 6.

If N = 4 and your goal is to find Five consecutive even integers, Then your five integers are: 4, 6, 8, 10, and 12.

If N = 4 is what you got, and you want to find Three consecutive multiples of 4, Then your three integers are: 4, 8, and 12.

If N = 4 is the solution of your equation, and you want the second number in the list of consecutive multiples of 4, then your answer is: 8.

And if the answer N = 4 is what you get when you solve an equation to find four consecutive odd integers, then you’ve done something wrong — or the problem was written incorrectly. The number 4 isn’t odd. When the equations are set up and solved correctly, you get a number belonging to the category you want.

Writing up sums and solving them

Consecutive integer problems have a common theme: Take a list of consecutive integers, perform a certain operation, and you get a numerical result. When writing the equations needed to solve these problems, you use a fairly common pattern: Let N Represent the first number in the list, let N + D (where D Is the common difference) be the second number, let N + 2D Be the third number, and so on. The most common operation performed in these problems is addition — so add ‘em up.

The Problem: The sum of three consecutive integers is 45. What are the integers?

Write the three integers symbolically as: N, n + 1, and N + 2. Now write the three numbers all added together and set them equal to 45: N + (n + 1) + (n + 2) = 45. Drop the parentheses, and combine the terms on the left to get 3n + 3 = 45. Subtract 3 from each side, and 3n = 42. Divide each side by 3, and you

Get N = 14. The three consecutive integers with a sum of 45 are 14, 15, and 16.

The Problem: The sum of four consecutive integers is 38. What is the largest number?

Write the four integers symbolically as: N, n + 1, N + 2, and N + 3 . Add them up, and let the sum be equal to 38. N + (n + 1) + (n + 2) + (n + 3) = 38. Simplifying the terms on the left, the equation becomes 4N + 6 = 38. Subtract 6 from each side to get 4N = 32. Dividing each side by 4, you get N = 8. So the four terms are 8, 9, 10, and 11. The largest of the four consecutive integers is 11.

The Problem: The sum of eight consecutive integers is 4. What is their product?

You may think that there’s been some error here. How can eight consecutive integers add up to a number smaller than the number of integers? The answer to that question is: "Negative numbers." Keep that in mind as you answer the question posed in this problem.

Write the eight consecutive integers as N, n + 1, N + 2, N + 3, N + 4, N + 5, N + 6, and N + 7. Adding them all up, you get N + (n + 1) + (n + 2) + (n + 3) + (n + 4) + (n + 5) + (n + 6) + (n + 7) = 4. Simplifying, the equation becomes 8N + 28 = 4. Subtract 28 from each side, and the equation 8N = -24 is ready to have each side divided by 8. With N = -3, the eight terms are: -3, -2, -1, 0, 1, 2, 3, and 4. Multiplying all the terms together, you get that the product is 0, because one of the terms is 0.

Looking At Consecutive Multiples

Consecutive integers are lists of integers that have a common difference between the terms. Odd and even numbers have a common difference of two between each term. Multiples of three have a common difference of three between the consecutive terms, and so on.

Working with evens and odds

The hardest part of doing problems involving consecutive even and consecutive odd integers is accepting the fact that the expressions for consecutive even and consecutive terms are the same. If you say that N, n + 2, N + 4, and N + 6 are four consecutive even numbers, and that N = 10, then you just add 2, 4, and 6 to the 10 and get 12, 14, and 16. Now, take the same four numbers, N, n + 2, N + 4, and N + 6, and say that they’re all odd numbers. Letting N = 11 this time, you see that when you add 2 to 11 you get 13. Adding 4 gives you 15, and adding 6 gives you 17. It’s the value of N That sets the pattern.

The Problem: The sum of four consecutive odd numbers is 64. What are the numbers?

£,?LAiV First write the four numbers symbolically as N, n + 2, N + 4, and N + 6. Adding them together and setting the sum equal to 64, N + (n + 2) + (n + 4) + (n + 6) = 64. Simplifying the equation, you get 4N + 12 = 64. Subtract 12 from each side, giving you 4N = 52. Dividing each side of the equation by 4, N = 13. The other odd numbers are 15, 17, and 19.

The Problem: The sum of five consecutive even integers is 100. What is the difference between the largest and the smallest of these integers?

^VLA/V The five consecutive even integers are written as N, n + 2, N + 4, N + 6, and N + 8. Adding them together and setting the sum equal to 100, N + (n + 2) + (n + 4) + (n + 6) + (n + = 100. Simplifying the equation, 5N + 20 = 100. Subtract 20 from each side to get 5N = 80. Dividing each side by 5, you get that N = 16. Rather than list the five integers, just find the largest even integer in the list by using N + 8 where N = 16. Since 16 + 8 = 24, you know that the largest integer is 24 and the smallest is 16. The difference between the two integers is 8.

In the previous problem, you really didn’t need to find the even numbers in the list. If you have five consecutive even integers, they’re always N, n + 2, N + 4, N + 6, and N + 8, no matter what N Is. The difference between the largest and the smallest is (n + – N = 8.

Expanding to larger multiples

Pick a number, and you can make a list of integers that are multiples of that number. You can also make a symbolic list (using math symbols) of integers that are multiples of a number by starting with N And adding the number repeatedly. For example, a list of six consecutive multiples of 7, starting with 63 is: 63, 70, 77, 84, 91, 98. Symbolically, a list of any six multiples of 7 is: N, n + 7, N + 14, N + 21, N + 28, N + 35.

The Problem: The sum of four consecutive multiples of 5 is 210. What is the largest of the four multiples?

Write the four integers as N, n + 5, N + 10, and N + 15. Add them up and set the sum equal to 210. The equation is: N + (n + 5) + (n + 10) + (n + 15) = 210. Simplifying on the left, the equation becomes 4N + 30 = 210. Subtract 30 from each side, and the equation 4N = 180 is solved by dividing both sides by 4.

The value of N = 45. The original question asks for the largest of these consecutive integers. From the list of integers, the largest is N + 15. So, if N = 45, then N + 15 = 45 + 15 = 60. Or, another approach is to just write the four multiples of 5, starting with 45: 45, 50, 55, 60. That’s a nice way to check your work.

The Problem: The sum of five consecutive multiples of 3 is 15. What is the product of the largest two of the numbers?

Write the five integers as N, n + 3, N + 6, N + 9, and N + 12. Adding them together and setting the sum equal to 15, you get 5n + 30 = 15. Subtract 30 from each side and divide by 5 to get 5n = -15 and N = -3. The two largest numbers in the list of consecutive integers are found with N + 9 and N + 12, making them equal to 6 and 9. The product of 6 and 9 is 54.

Operating on consecutive integers

Consecutive integers can be added, subtracted, multiplied, and divided. Many interesting problems come from performing various operations on the integers and not just limiting yourself to addition. Also, consecutive integers can be squared and cubed and more, before being combined in some fashion — just to mix things up a bit.

Doing more than addition to consecutive integers

One nice thing about just adding consecutive integers is that you always have a linear equation to solve. The equations formed from adding consecutive integers all seem to boil down to something in the form Ax + B = C, Which is solved by subtracting and dividing. But, on the other hand, isn’t it nice that some other types of equations, such as quadratic equations, are created as soon as other operations arrive on the scene?

The Problem: The product of two consecutive integers is 20. What are the integers?

Write the two consecutive integers as N And N + 1. Multiply them together and set the equation equal to 20. You get a quadratic equation that’s solved by setting the equation equal to 0, factoring, and solving for the numbers.

The quadratic equation gives you two different answers. When N = 4, you get the two consecutive integers 4 and 5. The product of 4 and 5 is, indeed, 20. But what about the solution N = -5? If N = -5, then N + 1 = -5 + 1 = -4. The product of -5 and -4 is also 20. This problem has two different solutions. As long as you’re happy with negative integers, too, then you accept both sets of answers.

The Problem: The product of two Positive Consecutive odd integers is 143. What are the integers?

The product of the two consecutive odd integers is written as N (n + 2). Set that product equal to 143, set the equation equal to 0, and solve the quadratic equation that results.

The solution N = 11 is used to find the other odd integer by adding 2. The two consecutive integers are 11 and 13. You don’t bother with the solution N = -13, because the problem specifies that you’re to find positive integers.

Working with consecutive squares and cubes

The numbers 1, 2, 3, and 4 are four consecutive integers. The numbers 1, 4, 9, and 16 are four consecutive Squares Of those first four numbers. And the Cubes Of the same four numbers are 1, 8, 27, and 64. Squares and cubes of consecutive numbers are squared and cubed, respectively, Before Being added together or having some other operation performed upon them.

The Problem: The sum of the squares of two consecutive integers is 145. What are the numbers?

LAiV First write the two consecutive integers as N And N + 1. The squares of those two numbers are N2 And (n + 1)2. To write the equation, add the two squares together and set them equal to 145. Then do the squaring on the left, combine like terms, subtract 145 from each side to set the equation equal to 0, and solve the quadratic equation.

When N = 8, then N + 1 = 9. The sum of their squares is 64 + 81 = 145. The other solutions works as well. When N = -9, then N + 1 = -8. Their squares are 81 and 64 and also have a sum of 145.

The Problem: The difference between the cubes of two consecutive numbers is 127. What are the numbers?

First write the cubes of the numbers as N3 And (n + 1)3. Subtract the smaller number from the larger number and set the difference equal to 127. The equation you write is (n + 1)3 – n3 = 127. Now cube the binomial and simplify the terms on the left. The resulting equation is quadratic, which factors and yields two solutions.

(N + 1)3 – N3 = 127 N3 + 3n2 + 3n + 1 – N3 = 127 3n2 + 3n + 1 = 127 3n2 + 3n – 126 = 0 3 (N2 + N – 42) = 0 2 (N – 6)(N + 7) = 0

N – 6 = 0, N = 6 N + 7 = 0, N = -7

When N = 6, you get 6 and 7 whose cubes are 216 and 343, respectively. The difference between their cubes is 127. When N = -7, you get -7 and -6 whose cubes are -343 and -216. Subtracting -216 – (-343) you also get 127.

The expansion of the binomial (n + 1)3 results in a polynomial where the coefficients of the terms have a distinctive, symmetric pattern. Refer to the Cheat Sheet for some of the other powers of a binomial. You’ll see that the cube of (n + 1) has coefficients in the 1-3-3-1 pattern.

Finding Sums of Sequences of Integers

A Sequence Of numbers is a list of numbers created by a particular pattern or mathematical rule. An Arithmetic sequence Is a list of numbers in which there is a common difference between the consecutive numbers in the sequence. So consecutive integers are a special type of arithmetic sequence. The rule that allows you to add up any number of terms in an arithmetic sequence also lets you solve some problems involving the sums of consecutive integers.

Setting the stage for the sums

Before applying the rule for the sum of a certain number of terms in an arithmetic sequence, you need to be able to find the NTh term in a sequence when

Given the first term or find the N When given the first and last terms. Then you can apply the general rule for the sum of the terms.

Finding the nth term or finding the difference

The NTh term of an arithmetic sequence may be the fourth term or the tenth term or any number of term. You don’t want to have to list the first 99 terms in order to find the hundredth term, so the following rule comes in handy.

The nth term of an arithmetic sequence is an, which is equal to a1 + D(n - 1), where a1 is the first term and D Is the difference between each of the terms in the sequence.

You use the formula for the nth term of an arithmetic sequence, An = a1 + d(n -1), to find a particular term in the sequence.

The Problem: In the sequence of terms beginning with 7 and with a common difference of 5, what is the tenth term in that sequence?

Using the formula and replacing D With 5 and N With 10, a10 = 7 + 5(10 – 1) = 7 + 5(9) = 7 + 45 = 52. You can check this answer, because you only have ten terms to worry about. Here are the first ten terms of the sequence: 7, 12, 17, 22, 27, 32, 37, 42, 47, 52. Yes, the tenth term is 52. You want to be sure that the formula works when the sequence has 100 or 1,000 terms.

The Problem: If the first term of an arithmetic sequence is 3 and the 50th term is 297, then what is the common difference between the terms?

Using the formula for finding the NTh term, you fill in values for everything except the difference and then solve for that difference. The equation is written 297 = 3 + D(50 – 1). Simplifying on the right, the equation becomes 297 = 3 + 49d. Subtract 3 from each side for 294 = 49d. Dividing each side of the equation by 49, you get that D = 6.

Summing up the terms

The generalized formula for the sum of any number of terms of an arithmetic sequence allows you to add up the terms no matter where you start and where you stop in the sequence.

The sum of N Terms of an arithmetic sequence is equal to half N Times the sum of the first and last terms of the sequence.

N(a 1+ AN)

The Problem: Find the sum of the 16 terms in the sequence that start with the number 15 and end with the number 60.

16

Using the formula, and substituting in the values, S16 = (15 + 60) =

8 (75) = 600. Do you wonder what those terms are? You find them by using the formula for the nth term. Using 60 = 15 + d(16 – 1) which becomes 60 = 15 + 15d, 45 = 15d, or D = 3. So the numbers are 15, 18, 21, 24, . . . 57, 60. (Okay, I was curious about the numbers but not interested enough to write out all 16 of them.)

The Problem: If the sum of the ten terms in an arithmetic sequence is 1,135 and if the difference between the terms is 3, then what are the first and last terms?

^VLA* First, write the first and last terms in an expression that relates one to the other. The tenth term is a10 = a1 + 3(10 – 1) = a1 + 27, using the formula found in Finding the nth term or finding the difference. Now use the sum formula, replacing each letter in the formula with its equivalent.

S n = n (a 1+ An)

1,135 = 10 (A 1 + A 1 + 27) 1,135 = 5 (2a1+ 27)

Divide each side by 5 and solve for a1. When you find the first term, you add 27 to find the tenth term.

1,135_5 (2a1+ 27)

5 5

227 =2A1+27

200 = 2A1 100 = A1 127 = a10

Finding the sums of consecutive integers

In most consecutive integer problems, you’re given the sum of a certain number of integers and told to figure out what those integers were that gave you a particular sum. For a change of pace, in the problems in this section, it’s the sums that you compute. And, because the problems involve lots and lots of consecutive integers, you find the formulas to be very nice.

The Problem: Find the sum of the 20 consecutive multiples of 4 that begin with the number 60.

Strange book

There’s a book in the local library where the for- introduction, and the glossary comes before the ward comes after the epilogue, the end is in the title. What is the name of this book? first half of the book, the index comes before the

■Ajeuoijoip B p8||E0 S, i\:jaMsu\f

The formula for finding the sum of a list of consecutive integers requires that you have the first and last terms in the list. The multiples of 4 are all four units apart. You have an arithmetic sequence with terms the difference of which is 4. (You can find more on arithmetic sequences in "Setting the stage for the sums," earlier in this chapter.) So, to find the 20th term in the list of multiples of 4 that start with 60, use the formula an = a1 + D(n - 1), giving you that a20 = 60 + 4(20 – 1) = 60 + 4(19) = 60 + 76 = 136. Now, using the formula for the sum of the terms, and letting 60 be the first term and 136 be the 20th term,

S n = n _ a 1+ An) 20

S20 = ~2 (60 + 136) = 10 (196) = 1,960

Applying Consecutive Integers

Adding up lists of numbers is always a huge amount of fun — or not. It depends on what you like to do with your leisure time, I suppose. When formulas are available to make arithmetic computations easier and more accurate, you jump at the chance to use those formulas. Here I give you some applications of sums of consecutive integers.

Adding up building blocks

A child’s set of building blocks usually consists of many wooden cubes — all the same size and decorated with letters of the alphabet, numbers, animals, or different colors. A typical stacking exercise involves making a row of blocks, with each block touching the one next to it, and then making a row on

Top of that first row where each block in the second row straddles two blocks beneath it (it sits on the crack between the two blocks). In this way, as more rows are added, each row has one less block than the row it’s sitting on.

The Problem: Little Jimmy is stacking his cube-shaped blocks. His first row has 20 blocks, and each subsequent row has one block less than the one below. If there are 250 blocks in Jimmy’s set, will he have enough blocks to build a structure all the way up to one block on the top?

Find the sum of the consecutive integers 1, 2, 3, 4, 5, . . . 19, 20 by using the formula for the sum of consecutive integers. The first term is 1; the last term is 20; and the number of terms, N, Is 20. Compare that sum with the number of blocks to see if Jimmy will have enough blocks.

S n = n _ a 1+ An) 20

S 20 = -J – (1 + 20) = 10 (21) = 210

Jimmy has more than enough blocks. In fact, he’ll probably throw those extra blocks at his brother.

Finding enough seats

In most theaters, you have more seats in the rows toward the back and fewer seats in the rows up front as the room narrows toward the stage. Imagine having to clean up after a performance attended by a rowdy bunch of children who were all given huge bags of popcorn. You need to plan the amount of time needed to do the job.

The Problem: The theater at a local civic center has 13 seats in the first 3 rows, 15 seats in the next 3 rows, and the number of seats increases by 2 seats every third row until there are a total of 45 rows in the theater. How many seats are there?

^VLAAf Because the number of seats in a row is repeated three times, just figure out how many seats in one of each of those different rows and multiply the result by three. With 45 rows total, then there are 15 different numbers of seats in the rows (45 3 = 15). The first row has 13 seats, and the number of seats per row increases by 2, so the number of seats per row is: 13, 15, 17, 19. . . . The 15th different row has a1 + D(n - 1) = 13 + 2(15 – 1) = 41 seats. The sum of the 15 numbers starting with 13 and ending with 41 is:

15

S15 = Y (13 + 41)

= ^ (5427) = 15 (27) = 405

If there were just one of each of these 15 different rows, you’d have 405 seats. But there are 45 rows — 3 of each of those 15 different ones. Multiplying 405 by 3, you get 405 x 3 = 1,215 seats all together.

Laying bricks for a stairway

Stairways to monuments get a lot of wear and tear, so paving bricks are used, because they can take all the punishment of many feet over many years.

The Problem: A brick stairway to a monument has 40 steps. The first step has 250 bricks, and each successive step has 3 fewer bricks. How many bricks are in the last step, and how many total bricks are there in the stairway?

«*,VLA/V You use the formula for the nth term of a sequence of numbers letting the difference be -3. So the 40th step has a1 + d(n – 1) = 250 – 3(40 – 1) = 133 bricks. The sum of the bricks in the 40 steps, going from 250 bricks down

To 133 bricks, is ■4°(250 + 133) = 20 (383) = 7,660 bricks.

Chapter 13

The 5th Wave Bt am

"I sense ihai you’re becoming wore defensive and unapproachable lately."

93

In this part…

Sometimes it can seem as if no one understands your problem, but we do! These chapters give you CBT ammunition to surmount depression, obsessions, anxiety, and even unbridled rage. Read on to gain more control over your problems and really begin to realise recovery.