# Susita's Account

## Task

At the beginning of January, Susita had some money in her checking account. At the end of each month she deposits enough to double the amount currently in the account. However, she has a loan to pay off, requiring her to withdraw \$10 from the account monthly (immediately after her deposit).

- Assuming January is the first month, write an equation that describes the amount of money in Susitaâ€™s account at the end of the $n^{\rm th}$ month, $S(n)$, in terms of the amount of money in Susitaâ€™s account at the end of the $(n-1)^{\rm th}$ month, $S(n-1)$.
- At the end of May, Susita had \$2 left in the account. How much did she have at the end of January?

## IM Commentary

This task asks students to determine a recursive process from a context. Students who study computer programming will make regular use of recursive processes.

Adapted from Math Forum POW#1720, http://mathforum.org/fe

## Solution

- $S(n) = 2S(n-1) -10$, $n \ge 2$
- We are given $S(5) = 2$. And by a), we know that $S(5)= 2S(4)-10$ so we have $2=2S(4)-10$. Solving for $S(4)$ yields $S(4)= 6$. Similarly, $S(4)= 2S(3)-10$ so $S(3) = 8$. $S(3)= 2S(2)-10$ so $S(2) = 9$. $S(2)= 2S(1)-10$ so $S(1) = 9.50$. So Susita had \$9.50 in her account in early January.

## Susita's Account

At the beginning of January, Susita had some money in her checking account. At the end of each month she deposits enough to double the amount currently in the account. However, she has a loan to pay off, requiring her to withdraw \$10 from the account monthly (immediately after her deposit).

- Assuming January is the first month, write an equation that describes the amount of money in Susitaâ€™s account at the end of the $n^{\rm th}$ month, $S(n)$, in terms of the amount of money in Susitaâ€™s account at the end of the $(n-1)^{\rm th}$ month, $S(n-1)$.
- At the end of May, Susita had \$2 left in the account. How much did she have at the end of January?