Rule for Summing a Finite Geometric Series.

Question

What is the rule?

$$ \sum_{i=1}^{N}6^i $$

and

$$ \sum_{i=5}^{N}6^i $$

Mostly interested in #1 and the summation rules for c^x. But if you feel like doing my HW go ahead :)

Answer 1

Here is one approach to summing geometric series that I have seen:

$$ S = \sum_{i=1}^5 r^i = r + r^2 + r^3 + r^4 + r^5.$$ $$ rS = r^2 + r^3 + r^4 + r^5 + r^6.$$ $$ S - rS = S(1-r) = r - r^6.$$

So

$$ S = \frac{r - r^6}{1 - r}.$$

Notes: (a) I used the same notation as the Wikipedia article linked above, in case you want to look at both. (b) If $r = 1/2,$ something similar can be used to find $\sum_{i=1}^\infty (1/2)^i = 1.$

Answer 2

This is a famous use of telescoping series. We can write down the general case as:

$$S_N=\sum_{n=n_0}^Nr^n$$

Or, alternatively:

$$S_N=r^{n_0}+r^{n_0+1}+...+r^{N-1}+r^N$$

The "trick" is to multiply both sides by $(1-r)$. Many of the terms on the left will cancel, and you should be able to obtain a simple expression for $S_N$.