# Series

### Topics

## Introduction to Series - At A Glance:

Since arithmetic and geometric series are common menu items that everyone loves, they show up a lot in word problems. Unlike hotdogs and burgers, the word problems usually provide some individual quantities. They ask you to compute a related quantity or to combine the quantities.

### Sample Problem

Jenna makes a New Year's resolution to save her pennies. She saves 1 penny on January 1st, 3 pennies on January 2nd, 5 pennies on January 3rd, and so on. How many pennies does she save on the last day of January?

Answer.

On the *n*th day of January Jenna saves *a _{n}* pennies, where

*a*_{1} = 1

*a*_{2} = 3

*a*_{3} = 5

These are the terms of an arithmetic series with *a*_{1} = 1 and *d* = 2. Since January has 31 days, we are looking for the value of *a*_{31}. Using the magical formula for the *n*th term of an arithmetic series,

*a*_{31} = *a*_{1} + (31 – 1)*d*

= 1 + 30(2)

= 61.

On the last day of January Jenna saves 61 pennies.

*If a problem involves quantities that are changing via addition or subtraction, there's an arithmetic series buried in that mountain of words somewhere. If the quantities are changing via multiplication or division, there's a geometric series lurking.*

Beyond that, our best advice for series word problems is to write out enough terms so that you can see the pattern. In the example with Liam and his piggy bank, it would be perfectly reasonable to write out some sort of table like this:

Then we could use the table to figure out the formula for the number of pennies *a _{n}* on day

*n*.

**Useful Trick:** When trying to calculate general formulas for the terms *a _{n}* of a sequence or series, sometimes it's more helpful to write

*a*as a series than it is to simplify

_{n}*a*all the way to a number. We can stick the series for

_{n}*a*into the formula for

_{n}*a*

_{n + 1}, and simplify just enough so that we still have a series. This can make it easier to spot the pattern.

This trick will be useful again in a few exercises, don't forget about it.

#### Example 1

On the first day of February Liam feeds one penny to his piggy bank. On each subsequent day he feeds his piggy bank twice as many pennies as he did the day before. (a) How much money is in Liam's piggy bank at the end of the first week of February? (b) How much money is in Liam's piggy bank at the end of February? Assume it's not a leap year. |

#### Example 2

On January 1st Kendra puts $100 into her bank account. On the first of each month after that she deposits an additional $100. Unfortunately, during the middle of the month she always has to withdraw 75% of her money to pay bills. (a) How much money is in Kendra's account after her March deposit? (b) How much money is in Kendra's account after the |

#### Exercise 1

Once a week Mrs. Baker makes sugar cookies. The first week she makes the recipe, she uses the full 2 cups of sugar called for. Each week after that, she reduces the amount of sugar by one third.

(a) How much sugar does she use for the cookies on the fifth week?

(b) How much sugar does she use for cookies over half a year?

(c) If Mrs. Baker became immortal and baked cookies every week for all eternity, how much sugar would she use?

#### Exercise 2

Mr. Vold is a sadistic teacher who likes writing lots of exam questions. He usually starts out the semester with only 10 questions on the first exam, but for each subsequent exam he writes one and a half as many questions as were on the previous exam! Since there's no such thing as half a question and Mr. Vold likes writing questions, round your answers up to the next integer.

(a) How many questions are on the second exam of the semester?

(b) How many questions are on the third exam of the semester?

(c) How many questions are on the fifth exam of the semester?

(d) If Mr. Vold wrote 20 exams in a semester, how many total exam questions would they have all together?

#### Exercise 3

Mo reads a lot of books. When he was five years old he read 4 books. Each year he reads three more than twice the number of books he read the previous year.

(a) How many books does Mo read when he's six years old?

(b) How many books does Mo read when he's eight years old?

(c) Give a formula for the number of books Mo reads when he's 5 + *i* years old. It's ok if there's a summation sign in your formula.

(hint: When finding the answers to (a) and (b), don't simplify too much).

(d) Use your formula from (c) to calculate how many books Mo reads when he's fifteen years old.

Check your answer by computing the same number without using the formula.

#### Exercise 4

On January 1st Komi puts $100 into his bank account. On the first of each month after that he deposits an additional $10.

(a) How much money is in Komi's account at the end of February?

(b) How much money is in Komi's account at the end of March?

(c) How much money is in Komi's account after *n* months?

#### Exercise 5

On January 1st Kendra puts $100 into her bank account. On the first of each month after that she deposits an additional $100. Unfortunately, during the middle of the month she always has to withdraw 75% of her money to pay bills.

(a) How much money is in Kendra's account at the end of March?

(b) How much money is in Kendra's account at the end of *n* months? Write your answer in closed form.

#### Exercise 6

Lizette decides that starting in January she will deposit $50 into her bank account at the start of each month. Her account earns 0.25% interest per month. Interest is calculated at the end of each month. Truncate answers to two decimal places.

(a) In the middle of February, how much money is in Lizette's account?

(b) In the middle of March, how much money is in Lizette's account?

(c) In the middle of the *n*th month (where January is the 1st month, February is the 2nd month, etc.), how much money is in Lizette's account? Give your answer in closed form.

(d) In the middle of December, how much money is in Lizette's account?