Answer
True statement
Work Step by Step
Sequence is $2,6,18,54,\ldots $
The general term or ${{n}^{th}}$ term of the geometric sequence is ${{a}_{n}}$.
and ${{a}_{n}}={{a}_{1}}{{r}^{n-1}}$
Where, ${{a}_{1}}$is the first term and $r$ is the factor between the terms (common ratio).
Here ${{a}_{1}}=2$
And $r=\frac{{{a}_{n+1}}}{{{a}_{n}}}$
Find the value of $r$, put $n=1$ in $r=\frac{{{a}_{n+1}}}{{{a}_{n}}}$
$\begin{align}
& r=\frac{{{a}_{n+1}}}{{{a}_{n}}} \\
& =\frac{{{a}_{1+1}}}{{{a}_{1}}} \\
& =\frac{{{a}_{2}}}{{{a}_{1}}}
\end{align}$
Put ${{a}_{1}}=2$ and ${{a}_{2}}=6$
$\begin{align}
& r=\frac{6}{2} \\
& =3
\end{align}$
Next term of the given sequence $2,6,18,54,\ldots $is ${{5}^{th}}$ term.
For the fifth term put $n=5$ in the equation ${{a}_{n}}={{a}_{1}}{{r}^{n-1}}$
$\begin{align}
& {{a}_{5}}={{a}_{1}}{{r}^{5-1}} \\
& =2\times {{3}^{4}} \\
& =2\times 81 \\
& =162
\end{align}$
The next term in the geometric sequence $2,6,18,54,\ldots $ is 162.
Thus, the given statement “The next term in the geometric sequence $2,6,18,54,\ldots $ is 162.” is true.
