# The first term of a geometric sequence is 4 and the multiplier, or ratio, is –2. What is the sum of the first 5 terms of the sequence?

Aug 4, 2018

${S}_{5} = 44$

#### Explanation:

$\text{the sum to n terms of a geometric sequence is}$

•color(white)(x)S_n=(a(r^n-1))/(r-1)

$\text{where a is the first term and r the common ratio}$

$\text{here "a=4" and } r = - 2$

${S}_{5} = \frac{4 \left({\left(- 2\right)}^{5} - 1\right)}{- 2 - 1}$

$\textcolor{w h i t e}{{S}_{5}} = \frac{4 \left(- 32 - 1\right)}{- 3} = \frac{- 132}{- 3} = 44$

$\text{Alternatively}$

$\text{listing the first five terms of the sequence}$

$4 , - 8 , 16 , - 32 , 64$

${S}_{5} = 4 - 8 + 16 - 32 + 64 = 44$

Aug 4, 2018

$= {S}_{5} = 44$

#### Explanation:

The sum of first $n$ term of geometric sequence is :

color(blue)(S_n=(a_1(1-r^n))/(1-r)

Where , ${a}_{1} =$ first term $\mathmr{and} r =$ common ratio.

We have , ${a}_{1} = 4 \mathmr{and} r = \left(- 2\right)$

So, the sum of first $5$ terms is$= {S}_{5} \to n = 5$

$\therefore {S}_{5} = \frac{4 \left(1 - {\left(- 2\right)}^{5}\right)}{1 - \left(- 2\right)}$

$\implies {S}_{5} = \frac{4 \left(1 - \left(- 32\right)\right)}{1 + 2}$

$\implies {S}_{5} = \frac{4 \left(1 + 32\right)}{3} = \frac{4 \times 33}{3} = 4 \times 11$

$\implies {S}_{5} = 44$