Answer
The provided statement is falseand if the sequence is \[2,6,18,54,\ldots \]common ratio is same the given sequence will be G.P.
It is known that the geometric sequence is;
\[a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}},\ldots \]
Where a is first term and r is common ratio,
\[\begin{align}
& r=\frac{{{a}_{2}}}{{{a}_{1}}} \\
& =\frac{{{a}_{3}}}{{{a}_{2}}} \\
& =\frac{{{a}_{3}}}{{{a}_{4}}}
\end{align}\]
Work Step by Step
In the provided sequence,\[{{a}_{1}}=2,{{a}_{2}}=6,{{a}_{3}}=24,{{a}_{4}}=120\],
\[\begin{align}
& {{r}_{1}}=\frac{6}{2} \\
& =3 \\
& {{r}_{2}}=\frac{24}{6} \\
& =4
\end{align}\]
And,
\[\begin{align}
& {{r}_{3}}=\frac{120}{24} \\
& =5
\end{align}\]
So,
\[\begin{align}
& {{r}_{1}}\ne {{r}_{2}} \\
& \ne {{r}_{3}}
\end{align}\]
Hence, the provided sequence is not geometric Sequence.
If the sequence is \[2,6,18,54,\ldots \]common ratio is same the provided sequence will be G.P.
Hence, the provided statement is false, if the sequence is \[2,6,18,54,\ldots \]common ratio is same the provided sequence will be G.P.
