Question

Find the common ratio of the geometric sequence: $-1,1,-1,1...........$
$$\begin{gathered} \text{a}\text{. 1} \\ \text{b}\text{. }-1 \\ \text{c}\text{. 2} \\ \text{d}\text{. }-2 \end{gathered}$$

Answer 1

Hint: - Use common ratio of a G.P $={\displaystyle \frac{a_{n+1}}{a_n}}$.
Given geometric series is $-1,1,-1,1...........$

As we know for a G.P say $\left(a_{1,}{a}_{2,}{a}_{3,}{a}_{4,}............a_n,a_{n+!}\right)$ the common ratio $\left(r\right)$ is written as
$r={\displaystyle \frac{a_2}{a_1}}={\displaystyle \frac{a_3}{a_2}}={\displaystyle \frac{a_4}{a_3}}=..........{\displaystyle \frac{a_{n+1}}{a_n}}$

Where $\left(n=1,2,3..............\right)$

So, the common ratio $\left(r\right)$for the given series is$$r={\displaystyle \frac{1}{-1}}={\displaystyle \frac{-1}{1}}={\displaystyle \frac{1}{-1}}=-1$$
Hence option (b) is correct.

Note: - In such types of questions the key concept we have to remember is that always remember the formula of common ratio of a G.P which is stated above, then simplify we will get the required common ratio of a G.P.