Question

How do you write a rule for the nth term of the geometric sequence given the two terms `a_2=36, a_4=576`?

Answer 1

See explanation.

Explanation:

For any geometric sequence `a_n`, and natural `n,k` `(n>k)` you can write that:

`a_n=a_k*q^(n-k)`

In this task we can write that:

`576=36q^2`

`q^2=16`

This leads to 2 possible values of `q`: `q_1=-4` and `q_2=4`

Now we can calculate the first term separately for `q=-4` and `q=4`

If `q=-4` then `a_1=36/(-4)=-9`

else if `q=4` then `a_1=36/4=9`

So this task has 2 solutions:

`a_n=(-9)*(-4)^(n-1)` or `a_n=9*4^(n-1)`