# How do you find S_n for the geometric series a_1=4, r=0.5, n=8?

Dec 6, 2016

${S}_{n} = 7.96875$

#### Explanation:

there are two equations to find ${S}_{n}$ of a geometric series.
if r is greater than 1, we use ${S}_{n} = a \cdot \frac{{r}^{n} - 1}{r - 1}$
and if r is less than 1, we use ${S}_{n} = a \cdot \frac{1 - {r}^{n}}{1 - n}$

in this question r is less than 1. so we use the second equation to find ${S}_{n}$

${S}_{n} = 4 \cdot \frac{1 - {0.5}^{8}}{1 - 0.5}$

therefore ${S}_{n} = 7.96875$

Dec 6, 2016

${S}_{8} = \textcolor{g r e e n}{7.96875}$

#### Explanation:

Given an initial value $\textcolor{red}{{a}_{1}}$, and a geometric increment of $\textcolor{b l u e}{n}$
the ${\textcolor{b r o w n}{n}}^{t h}$ term is given by the formula:
$\textcolor{w h i t e}{\text{XXX}} {S}_{\textcolor{b r o w n}{n}} = \textcolor{red}{{a}_{1}} \left(\frac{1 - {\textcolor{b l u e}{r}}^{\textcolor{b r o w n}{n}}}{1 - \textcolor{b l u e}{r}}\right)$

Using the given values (and a calculator)
$\textcolor{w h i t e}{\text{XXX}} {S}_{8} = 7.96875$