Question

Which of the following sequences are geometric sequences
(i) $5,10,15,20,....$
(ii) $0.15,0.015,0.0015,.....$
(iii) $\sqrt{7},\sqrt{21},3\sqrt{7},3\sqrt{21},.....$

Answer 1

(i) Considering the ratios of the consecutive terms, we see that $\frac{10}{5}\ne \frac{15}{10}$.
Thus, there is no common ratio. Hence it is not a geometric sequence.
(ii) We see that $\frac{0.015}{0.15}=\frac{0.0015}{0.015}=...=\frac{1}{10}$.
Since the common ratio is $\frac{1}{10}$, then given sequence is a geometric sequence.
(iii) Now, $\frac{\sqrt{21}}{\sqrt{7}}=\frac{3\sqrt{7}}{\sqrt{21}}=\frac{3\sqrt{21}}{3\sqrt{7}}=....=\sqrt{3}$. Thus, the common ratio is $\sqrt{3}$.
Therefore, the given sequence is a geometric sequence.