How do you find the sum of the finite geometric sequence of Sigma 8(-1/2)^i from i=0 to 25?

Dec 9, 2017

See below.

Explanation:

The sum of a geometric series is given by:

$a \left(\frac{1 - {r}^{n}}{1 - r}\right)$

Where $a$ is the first term, $r$ is the common ratio and $n$ is the nth term.

For $i = 0$ the first term is:

$8 {\left(- \frac{1}{2}\right)}^{i} = 8 {\left(- \frac{1}{2}\right)}^{0} = 8 \left(1\right) = 8$

Common ratio is $- \frac{1}{2}$

$\therefore$

$8 \left(\frac{1 - {\left(- \frac{1}{2}\right)}^{25}}{1 - \left(- \frac{1}{2}\right)}\right) = 8 \left(\frac{1 - \left(- \frac{1}{32} ^ 5\right)}{\frac{3}{2}}\right)$

$= \left(\frac{16 + \left(\frac{16}{32} ^ 5\right)}{3}\right) = 5.333333254$