Question

What is the recursive formula of the geometric sequence 1/2,3/4,9/8,27/16

Answer 1

Explanation:

A recursive geometric formula has the pattern:

`A_n=A_(n-1)*r` where r is the common ratio. Before we write the formula, we need to find the value for r. This r value has to be the same between each consecutive number in the series. Because it's geometric, r is multiplied in (whereas it's added in if the sequence is arithmetic). To get from 1/2 to 3/4 we find the r value in this way:

`(x)/(2)=(3)/(4)` where x/2 is the same as 1/2 x. Cross multiply to get 4x = 6 and x = 3/2. That means that if r is 3/2, then we can multiply every term in that sequence by 3/2 to get to the next term in line. 3/4 times 32 is 9/8. Therefore, r = 3/2 and the formula will be:

`A_n=A_(n-1)*(3)/(2)`