# What is the sum of the geometric sequence -3, 18, -108, … if there are 7 terms?

Aug 14, 2018

${S}_{7} = - 119973$

#### Explanation:

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

•color(white)(x)S_n=(a(r^(n-1)))/(r-1)

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

$a = - 3 \text{ and } r = \frac{- 108}{18} = \frac{18}{- 3} = - 6$

${S}_{7} = \frac{- 3 \left({\left(- 6\right)}^{7} - 1\right)}{- 6 - 1}$

$\textcolor{w h i t e}{\times} = \frac{- 3 \left(- 279936 - 1\right)}{- 7}$

$\textcolor{w h i t e}{\times} = \frac{- 3 \times - 279937}{- 7} = - 19973$

Aug 14, 2018

${S}_{7} = - 119973$

#### Explanation:

Here,

$- 3 , 18 , - 108 , \ldots \text{are in GP}$

Let ,first term $= {a}_{1} = - 3 \mathmr{and}$

common ratio $= r = \frac{- 108}{18} = \frac{18}{- 3} = - 6$

So, the sum of first n terms is:

${S}_{n} = \frac{{a}_{1} \left(1 - {r}^{n}\right)}{1 - r} , w h e r e , n = 7$

$: {S}_{7} = \frac{- 3 \left(1 - {\left(- 6\right)}^{7}\right)}{1 - \left(- 6\right)}$

$\therefore {S}_{7} = - \frac{3 \left(1 + 279936\right)}{7} = - \frac{3 \left(279937\right)}{7} = - 119973$

Aug 14, 2018

-119973

#### Explanation:

We can first see that the ratio between these is $- 6$. This means we have the sum of a geometric series, which we know is
${S}_{n} = {a}_{1} \cdot \frac{{r}^{n} - 1}{r - 1} = - 3 \cdot \frac{{\left(- 6\right)}^{7} - 1}{\left(- 6\right) - 1} = - 119973$