Question

A sequence has a common ratio of Three-halves and f(5) = 81. Which explicit formula represents the sequence?

Answer 1

f(x) = 16*(3/2)^(x-1)

Explanation:

right on edge

Answer 2

`f(n) = (32)/(3)((3)/(2))^n`

Explanation:

Given

`r = (3)/(2)`

`f(5) = 81`

Required

The geometric sequence

A geometric sequence is represented as:

`f(n) = ar^(n-1)`

Replace n with 5

`f(5) = ar^(5-1)`

`f(5) = ar^4`

Substitute values for f(5) and r

`81 = a* ((3)/(2))^4`

Open bracket

`81 = a* (81)/(16)`

Make a the subject

`a = (81 * 16)/(81)`

`a = 16`

So, the explicit function is:

`f(n) = ar^(n-1)`

`f(n) = 16 * ((3)/(2))^(n-1)`

Split

`f(n) = 16 * ((3)/(2))^n / ((3)/(2))`

Convert to multiplication

`f(n) = 16 * ((3)/(2))^n * (2)/(3)`

`f(n) = (32)/(3)((3)/(2))^n`