Question

Applications

4. Erin is traveling abroad this summer and would like to have a bit of spending cash while
she's overseas. She has 100 dollars already saved and she plans on saving 40 dollars a month.
(a) Fill out the table below for the amount of money she saves as a function of how many
months she has been saving.
m
m1 2 3 4
a(m) 140
(b) Give a recursive definition for the sequence a(m). Don't forget to give an initial value.

Answer 1

Part a)

`m \to \: a(m) \\ 1 \to \: 100 \\ 2 \to \: 140 \\ 3 \to \: 180 \\ 3\to \: 220 \\ 4 \to \: 260`

Part b)

`a_n=a_(n-1)+40 \\ a_1=100`

Explanation:

Part a)

Erina already $100 saved.

This means

`a_1=100`

She plans on saving 40 dollars a month.

This means that the common difference is

`d = 40`

`m \to \: a(m) \\ 1 \to \: 100 \\ 2 \to \: 140 \\ 3 \to \: 180 \\ 3\to \: 220 \\ 4 \to \: 260`

Part b)

The recursive formula for an arithmetic sequence is given as

`a_n=a_(n-1)+d`

The common difference is d=40 and the first term is

`a_1=100`

We substitute d=40 into the formula to get:

`a_n=a_(n-1)+40 \\ a_1=100`