Question

What is `s_n` of the geometric series with `a_1=4`, `a_n=256`, and `n=4`?

Answer 1

The sum is `340`.

Since the `nth` term is given as `256` but `n` is given as `4`, that means `256` is the `4th` term. But the `4th` term of a GP equals `ar^3`, where a is the first term and `r` is the common ratio of the GP. Dividing `256` by the first term (which is given as `4`) shows us that `r^3 = 256/4 = 64`.

If `r^3 = 64`, then the common ratio `r` must equal `4` as well. This gives us all the information we need to use the formula for the sum of a GP, `S = (a(r^n - 1))/(r - 1)`.

In this case, `S = (4(4^4 - 1))/(4 - 1) = (4(256 - 1))/3 = 1020/3 = 340`.