# How do you find the sum of the infinite geometric series 1, 5, -25, 125,…?

##### 1 Answer

This is not a geometric series.

If the first term is replaced with

#### Explanation:

If the sequence of numbers in the question is accurate then this is not a geometric series.

The ratios of successive pairs of terms are

It would be a geometric series if the first term was

#-1, 5, -25, 125,...#

The general term of this sequence is

Then assuming that was the intention, the sum of this geometric series does not converge.

The sum of an infinite geometric sequence with general term

In general:

#sum_(n=1)^oo a r^(n-1) = a/(1-r)#

provided

We find:

#(1-r) sum_(n=1)^N a r^(n-1)#

#= sum_(n=1)^N ar^(n-1) - r sum_(n=1)^N ar^(n-1)#

#= a + color(red)(cancel(color(black)(sum_(n=2)^N ar^(n-1)))) - color(red)(cancel(color(black)(sum_(n=2)^N ar^(n-1)))) - ar^N#

#= a(1-r^N)#

Dividing both ends by

#sum_(n=1)^N a r^(n-1) = (a(1-r^N))/(1-r)#

If

Hence:

#sum_(n=1)^oo a r^(n-1) = lim_(N->oo) sum_(n=1)^N a r^(n-1) = lim_(N->oo) (a(1-r^N))/(1-r) = a/(1-r)#