Question

What is the 8th term of the geometric sequence if `a_3 = 108` and `a_5 = 972`?

Answer 1

`26244`

Explanation:

In a geometric sequence is valid the following rule

`a_(i+1)=k*a_i`, where `i in NN`

So

`a_5=k*a_4=k*(k*a_3)`
`a_5=k^2*a_3`
`972=k^2*108` => `k^2=9` => `k=3`

By the same token

`a_8=k^(8-5)*a_5`
`a_8=k^3*a_5=3^3*972=27*972` => `a_8=26244`

Answer 2

`T_8 = ar^7 = 12 xx 3^7 = 26 244`

Explanation:

Before we can find an unknown term of a geometric sequence, we need to know the first term `(a)` and the common ratio `(r)`

Each term can be written as `T_n = ar^(n-1)`

Let's divide the two terms we have been given, their formulae and their values:

`(T_5)/(T_3) = (ar^4)/(ar^2) = 972/108`

The following happens: `(cancelar^4)/(cancelar^2) = 972/108`

Subtract the indices: `a^2 = 9 " " rArr r = 3`

In `T_3 = ar^2, if r = 3,` then `a(3^2) = 108`

`9a = 108 " "rArr a = 12`

Great! now we have `a and r` so we can find any term we want.

`T_8 = ar^7 = 12 xx 3^7 = 26 244`