Partial sum of geometric series

[realitybugll]

ok so i know how to calculate the partial sum of a geometric series.

But let's say i only want to calculate the sum of every other term, how would i do this?

example:

.5^0+.5^1+.5^2+...+.5^n = (.5^(n+1) - 1)/(.5-1)

but what equation can i use to get the sum of only these terms:

.5^1+.5^3+.5^5...+.5^n where n is odd

 

[arildno]

Hint:

See if you can re-write it as another geometric series, utilizing that an odd number n=2m+1, for m integer.

 

[realitybugll]

yea i actually need to do it for the even ones for my equation haha

ok so i think i got it, not going to write it all out because i;m busy, but thank you :)

 

[HallsofIvy]

And, of course, (.5)ⁿ, for n, even, n= 2m, is (.25)^m so you have just another geometric series.