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)#