Question

Victoria had $200 in her account at the end of one year. At the first of each subsequent year she deposits $15 into the account

and earns 2% interest on the new balance, compounded annually. Which recursive formula represents the total amount of
money in Victoria's account at the end of the nth year?
a, - 1.02( a -1 +15), aq - 215
an - 15+1.02a -1, 2, - 215
2,- 1.02(an-1 +15); a, - 200
an = 15+1.02am-11 an - 200

Answer 1

The amount of money in Victoria's account at the end of n-th year will be,

$
`(965 * (1.02)^((n -1)) - 765)`

Explanation:

The amount of money in Victoria's account at the end of n-th year will be,

$
`(200* (1.02)^((n-1)) + 15 * ((1.02)^((n-1)) + (1.02)^((n -2)) + (1.02)^((n-3)) + ........ + (1.02)))`

= $
`(200 * (1.02)^((n -1)) + 15 * (1.02 * ((1.02)^((n-1)) - 1))/(0.02))`

= $
`(200 * (1.02)^((n -1)) + 15 * 51 * ((1.02)^((n-1)) - 1))`

= $
`(965 * (1.02)^((n -1)) - 765)`

Since, the interest is compounded over that $ 200 (which was there in the account at the end of 1st year) for (n-1) years and for (n-1) years, (n-2) years, (n -3) years, ...... 1 year respectively over those $ 15 which are deposited in the account at the end of 1 , 2, 3, ...... (n-1) years.

Answer 2

`a_(n)=\left [ a\left ( n-1 \right )+15 \right ]*1.02\\`

Explanation:

1) A Recursive formula always makes reference to the previous term, written as a function.

2) From $200 to $215 there was a 7.5% growth, so since the question states that from the beginning of the second year there will be a regular growth.

3) Since it is 2% of interest compounded annually q=1+ 0.02.

`A=215(1+(0.02)/(1))^(1)\\A=219.3`

Or simply

215*1.02=219.3

4) We can write a Recursive formula this way:

`a(n)=215`

`a(n)=a(n-1)(1.02)`

5) But since we have a pattern, She will deposit $15 yearly then we must make a little adjustment then add $15. to that. Considering the first term to be 200

`a_(n)=\left [ a\left ( n-1 \right )+15 \right ]*1.02\\a_(1)=200`