Question

Given f(n) is a geometric sequence where f(3)=32 and f(9)=72, find f(16).

I think the answer may be 64/27, but I don't know for sure.

Answer 1

f(16)=162

Explanation:

-In a geometric sequence, the quotient between any tow consecutive numbers is called the common ratio,r:

`r=(a_n)/(a_(n-1))`

Given that f(3)=32 and f(9)=72, we can find the common ratio as below:

`r=(a_n)/(a_(n-1))\\\\\therefore (f(9))/(f(3))=r^6\\\\(72)/(32)=r^6\\\\r^6=2.25\\\\r=2.25^(1/6)`

#We substitute and use the same formula to find the nth term:

`a_n=ar^(n-1)`

let f(3) be a(the first term):

`a_n=ar^(n-1)\\\\a_(16)=32(2.25^(1/6))^(16-3-1)\\\\a_(16)=32(2.25^(1/6))^(12)\\\\\\=162`

Hence, f(16)=162