# Thread: geometic series

1. ## geometic series

A geometic series of n terms that can be written:
a + ar + ar^2 + ar^3 + ... + ar^n-1 where a is the first term and r is the common ratio. If I express this in sigma notation would the lower limit be k=1 and the upper limit be n and the sum be r^n-1? and how would I write a forumula for this? Could I write Sn= a/(1-r) when -1 < r < +1?

2. Originally Posted by kcsteven
A geometic series of n terms that can be written:
$\displaystyle a + ar + ar^2 + ar^3 + ... + ar^{n-1}$ where a is the first term and r is the common ratio. If I express this in sigma notation would the lower limit be $\displaystyle k = 1$ and the upper limit be $\displaystyle n$ and the sum be $\displaystyle r^{n-1}$? and how would I write a forumula for this? Could I write $\displaystyle S_n= \frac{a}{1-r}$ when $\displaystyle -1 < r < +1$?
You're very close with your Sigma notation, you left out the a

$\displaystyle a + ar + ar^2 + ar^3 + ... + ar^{n-1} = \sum_{k=1}^{n} ar^{k-1} = a \sum_{k=1}^{n} r^{k-1}$

However, I'd probably write this, they're equivalent

$\displaystyle a + ar + ar^2 + ar^3 + ... + ar^{n-1} = \sum_{k=0}^{n-1} ar^k = a \sum_{k=0}^{n-1} r^k$

As for $\displaystyle S_n= \frac{a}{1-r}$, that is in fact the limiting sum of a geometric sequence for $\displaystyle |r|< 1$, so yes, you're correct in that statement.

Hope that helped