Question

Find out which of the following sequences are geometric sequences. For those geometric sequences, find the common ratio.
(i) $0.12,0.24,0.48,....$
(ii) $0.004,0.02,0.1,....$
(iii) $\frac{1}{2},\frac{1}{3},\frac{2}{9},\frac{4}{27},.....$
(iv) $12,1,\frac{1}{12},.....$
(v) $\sqrt{2},\frac{1}{\sqrt{2}},\frac{1}{2\sqrt{2}},....$
(vi) $4,-2,-1,-\frac{1}{2},......$

Answer 1

We need to check for Geometric sequences from the given series.

Suppose in G.P. three consecutive terms are $a,b,c$.

These three terms will satisfy the condition $b^2=a\times c$

And the common ratio $r$ will be given by $=\frac{b}{a}$

Solving using these equations we find that

$\left(i\right)$ is a G.P. with common ratio $2$

$\left(ii\right)$ is a G.P. with common ratio $5$

$\left(iv\right)$ is a G.P. with common ratio $\frac{1}{12}$

$\left(v\right)$ is a G.P. with common ratio $\frac{1}{2}$