Question

Find the sum of the first $20$terms of the geometric sequence$6,18,54,162,486,1,458,\dots$

Short answer

Sum of the first $20$terms of the geometric sequence is $\begin{array}{rcl}& & -3\left(1-3^{20}\right)\end{array}$.

Step-by-step solution

Step 1. Given information

The geometric sequence is$6,18,54,162,486,1,458,...$

Step 2. Find the first term and the common ratio.

Consider the geometric sequence $6,18,54,162,486,1,458,...$

Let $a,r$denote the first term and the common ratio respectively.

$\begin{array}{rcl}a& =& 6\\ r& =& \frac{18}{6}\\ & =& 3\end{array}$

Step 3. Find the sum of the first 20 terms of the geometric sequence.

Put $a=6,r=3,n=20$in $S_n=\frac{a\left(1-r^n\right)}{1-r}$

$\begin{array}{rcl}{S}_{20}& =& \frac{6\left(1-3^{20}\right)}{1-3}\\ & =& -3\left(1-3^{20}\right)\end{array}$