Answer
The provided statement: A geometric series \[a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}},\ldots \] is True.
Work Step by Step
If a sequence is geometric, we can write as many terms as we want by repeatedly multiplying by the common ratio.
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;
\[\begin{align}
& r=\frac{{{a}_{2}}}{{{a}_{1}}} \\
& =\frac{{{a}_{3}}}{{{a}_{2}}} \\
& =\frac{{{a}_{3}}}{{{a}_{4}}}
\end{align}\]
For example, if a sequence is:\[2,4,8,16,32,.\ldots \]
Where common ratio,\[r=\frac{{{a}_{2}}}{{{a}_{1}}}=2\]
\[\begin{align}
& {{a}_{1}}=2 \\
& {{a}_{2}}=2\times 2=4 \\
& {{a}_{3}}=4\times 2=8 \\
& {{a}_{4}}=8\times 2=16 \\
\end{align}\]
Thus, repeatedly multiply by the common ratio to make a geometric sequence.
Hence, the given statement is true.
