Question

What is the sum of a 7–term geometric series if the first term is –11, the last term is –171,875, and the common ratio is –5?

Answer 1

`-143231`

Explanation:

The general term of a geometric series can be described by the formula:

`a_n = ar^(n-1)`

where the initial term is `a` and common ratio `r`.

Then we find:

`(1-r) sum_(n=1)^N a_n`

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

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

`= a(1-r^N)`

So dividing both ends by `(1-r)` we find:

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

In our example, `N=7`, `a=-11` and `r=-5`

So:

`sum_(n=1)^7 (-11)(-5)^(n-1)`

`=((-11)(1-(-5)^7))/(1-(-5))`

`=((-11)(1-(-78125)))/6`

`=((-11)(78126))/6`

`=-11*13021`

`=-143231`