Question

How do you find the sum of the first $7$ terms of a geometric sequence if the first term is $-1$ and the common ratio is $-3$?

Answer 1

Find the sum of the first $7$ terms of a geometric sequence.

Since the first term $a$ is $-1$ and the common ratio $r$ is $-3$.

The formula for the sum of $n$terms of a geometric sequence is $S_n=a\left(\frac{1-r^n}{1-r}\right)$.

Find the first $7$ terms of a geometric sequence.

$S_7=a\left(\frac{1-r^7}{1-r}\right)$

Substitute $a=-1$and $r=-3$ in the above formula:

$\begin{array}{rcl}{S}_7& =& \left(-1\right)\left\{\frac{1-{\left(-3\right)}^7}{1-\left(-3\right)}\right\}\\ & \Rightarrow & S_7=-\left\{\frac{1-\left(-2187\right)}{1+3}\right\}\\ & \Rightarrow & S_7=-\left\{\frac{1+2187}{4}\right\}\\ & \Rightarrow & S_7=-\left\{\frac{2188}{4}\right\}\\ & \Rightarrow & S_7=-547\end{array}$

Therefore, the sum of first $7$ terms of a geometric sequence is $\begin{array}{rcl}{\mathit{S}}_{\mathbf{7}}& \mathbf{=}& \mathbf{-}\mathbf{547}\end{array}$.