Question

The third through the fifth terms of a geometric sequence are given by a3 = 18, a4 = 27 , and a5 = 81/2

Answer 1

Let a be the first term in the sequence. If r is the ratio between consecutive terms, then the second term is ar, the third term is ar ^2, the fourth is ar ^3, and so on, up to the n-th term ar ^(n - 1).

So the third, fourth, and fifth terms are such that

ar ^2 = 18

ar ^3 = 27

ar ^4 = 81/2

Solve for a and r :

(ar ^3) / (ar ^2) = 27/18   =>   r = 3/2

ar ^2 = a (3/2)^2 = 9/4 a = 18   =>   a = 8

Then the n-th term in the sequence is

ar ^(n - 1) = 9 (3/2)^(n - 1)

You can rewrite this by first rewriting 9 = 3^2, then

9 (3/2)^(n - 1) = 3^2 * 3^(n - 1) / 2^(n - 1) = 3^(n + 1)/2^(n - 1)