Question

If $x_1,x_2,x_3,x_4$ are in G.P then its common ratio is,

• A. ${\left(\frac{ar}{cp}\right)}^{\frac{1}{4}}$
• B. ${\left(\frac{cr}{ap}\right)}^{\frac{1}{3}}$
• C. ${\left(\frac{cr}{ap}\right)}^{\frac{1}{2}}$
• D. ${\left(\frac{ap}{bq}\right)}^{\frac{1}{2}}$

Answer 1

The correct option is A ${\left(\frac{ar}{cp}\right)}^{\frac{1}{4}}$

Given: $x_1,x_2,x_3,x_4$ are in G.P then

$x_2=x_1{r}^{\prime },x_3=x_1{r}_{\prime }^2,x_4=x_1{r}_{\prime }^3$

$x_1{x}_2=x_1^2{r}^{\prime }=\frac{c}{a}$ ...$\left(1\right)$

$x_3{x}_4=x_1^2{r}^{{\prime }^5}=\frac{r}{p}$ ...$\left(2\right)$

Dividing eqns$\left(2\right)$ by $\left(1\right)$ we get

$\frac{r^{{\prime }^5}}{r^{\prime }}=\frac{\frac{r}{p}}{\frac{c}{a}}$

$\therefore r^{{\prime }^4}=\frac{ra}{pc}$

$\therefore r^{\prime }={\left(\frac{ra}{pc}\right)}^{\frac{1}{4}}$

Hence option $A$ is the answer