# GeometricSums

ClassS04CS141 | recent changes | Preferences

### Geometric sums

A geometric sum is a sum in which each term is a constant factor larger than the previous term. (Or each term can be a constant factor smaller than the previous term.) For example, here is a common geometric sum:

1 + 2 + 4 + 8 + … + 2n = 2n+1 - 1.
And, more generally,
1 + x + x2 + x3 + … + xn = (xn+1 - 1)/(x-1) unless x=1.
Using sum notation:
i=0..n xi = (xn+1-1)/(x-1).

(We can prove this by multiplying both sides by (x-1) and simplifying.)

For analysis of algorithms, the important thing is that, for any x≠ 1, any geometric sum is proportional to its largest term. For example, for x> 1,

i=0..n xi = Θ(xn).

In words, a geometric sum is proportional to its largest term.

This fact is useful in analyzing algorithms, especially divide-and-concur algorithms.

Note that this is not true for all kinds of sums. For example, 1+2+...+n = Θ(n2).

It does hold for infinite sums too: 1 + 1/2 + 1/4 + ... + 1/2i + ... = i=0..∞ 1/2i = Θ(1).

### references:

ClassS04CS141 | recent changes | Preferences