Sum or mean of exponential numbers series

Question

Let's say i have number series of a power of 3.

3^1, 3^2, 3^3
(3, 9, 27)

How do I calculate the mean (without computing the SUM) or how do I find the sum without actually summing it?

With linear series it is easy, because the mean is the middle number. But I wonder how to do it with exponential number series. I am afraid I am unable to figure it out.

I only noticed that in case linear series the number of both sides of middle number are +x -x, and in case of exponential serie it is /x *x (1,2,3 = -1, +1 on sides of 2) and (3,9,27 is /3 and *3 on sides of 3) so there must be some rule in it to find the mean or sum too.

Second thing i figured out is that i can at least find the sum of powers, because it is the linear series too. So it is easy to get the sum of powers, but you can't do then like 3^6 if the sum of powers of (3, 9, 27) (powers - 1,2,3) is 6.

Motivation: if i will be able to find this, i am able to compute the future costs including economic inflation (sum of expenses for each year) where inflation is the "power" and base number are the expenses, and sum is the total expenses.

Answer 1

For |a|>1, sum of the series is $$ a +a^2+a^3 + \ldots +a^n = a \frac{a^n-1}{a-1}. $$ For |a|<1 sum is $(a(1-a^n))/(1-a)$

Answer 2

Let $S:=3+3^2+3^3$. Observe that $3S=3^2+3^3+3^4$, which is $S$ minus its first term, plus the next term in the geometric progression. Hence

$$3S=S-3+3^4$$

and

$$S=\frac{3^4-3}2.$$

This easily generalizes to any base and any number of terms: it is the half-difference of the afterlast term and the first over the base minus one.