## Thinking Mathematically (6th Edition)

The provided statement: A geometric series $a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}},\ldots$ is True.
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.