Question

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

Answer 1

Answer:

$f(n) = \frac{32}{3}(\frac{3}{2})^n$

Step-by-step explanation:

Given

$r = \frac{3}{2}$

$f(5) = 81$

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

$81 = a* (\frac{3}{2})^4$

Open bracket

$81 = a* \frac{81}{16}$

Make a the subject

$a = \frac{81 * 16}{81}$

$a = 16$

So, the explicit function is:

$f(n) = ar^{n-1}$

$f(n) = 16 * (\frac{3}{2})^{n-1}$

Split

$f(n) = 16 * (\frac{3}{2})^n \div (\frac{3}{2})$

Convert to multiplication

$f(n) = 16 * (\frac{3}{2})^n * \frac{2}{3}$

$f(n) = \frac{32}{3}(\frac{3}{2})^n$