Question

How do you find the sum of the finite geometric sequence of `sum_(i=0)^10 5(-1/3)^(i-1)`?

Answer 1

`S = 5(-1/3)^{-1}( {1 - (-1/3)^{11})/(1 - (-1/3)) ) = -221435/19683`

Explanation:

The `n` term finite geometric series that starts with 1 has a straightforward sum:

`S_n = sum_{k=0}^{n-1} r^k = {1 - r^n}/{1-r}`

For the general geometric series we multiply by the first term. Our series has 11 terms.

`S = sum_{i=0}^{10} 5 (-1/3) ^(i-1)`

`S = 5(-1/3)^{-1}( {1 - (-1/3)^{11})/(1 - (-1/3)) )`

`S = -221435/19683`