Question

yui makes a list of the balances in her savings account at the end of each month. she notices that each month’s total is 5% greater than the previous month’s total. she writes a recursive formula to describe the account balances.

Answer 1

`T_(n)=x(1.05)^(n-1)`

Explanation:

Yui makes a list of balances in her savings account at the end of each month.

Each month's total is 5% greater than the previous month's total.

Let the first month's saving is $x.

Then next month's saving will be = x +0.05x

= (1.05x)

3rd month's saving = 1.05x + 5% of (1.05x)

= 1.05x + 0.0525x = 1.1025x

Now the sequence is x, 1.05x, 1.1025x .......

It's a geometric sequence with first term as x and common ratio = 1.05

Explicit formula of a geometric sequence =
`ar^(4-1)`

Where a = first term = x

r = common ratio = 1.05

x = number of term

So recursive formula will be
`T_(n)=x(1.05)^(n-1)`

Answer 2

The recursive formula is f(n + 1) = 1.05f(n)