# Series

### Topics

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.