Question

How do you find `a_7` for the geometric sequence `729, -243,81,...`?

Answer 1

`a_7=1`

Explanation:

We should first find a general formula for the geometric sequence, which will be in the form

`a_n=a(r)^(n-1), r>=1, r` is the common ratio between terms, `a` is the first term.

We see `a=729,` to determine `r,` simply divide one of the terms by the term before it -- we're told the sequence is geometric, so we don't need to test all of the terms for the ratio, as we already know they'll share a common ratio.

`r=-243/729=-1/3`

So, we obtain

`a_n=729(-1/3)^(n-1)`

To find `a_7,` just plug in `n=7:`

`a_7=729(-1/3)^(7-1)=729(-1/3)^6=1`