Question

If the $pth,qth$ and $rth$ terms of a G.P are $a,b,c$, then ${\left(\frac{c}{b}\right)}^p{\left(\frac{b}{a}\right)}^r{\left(\frac{a}{c}\right)}^q$ is equal to

• A. $1$
• B. $a^p{b}^q{c}^r$
• C. $a^q{b}^r{c}^p$
• D. $a^r{b}^p{c}^q$

Answer 1

The correct option is A $1$

General term of G.P $\Rightarrow A{R}^{n-1}$

$\therefore a=A{R}^{p-1}$

$b=A{R}^{q-1}$

$c=A{R}^{r-1}$

$\therefore \left(\frac{c}{b}{)}^p\right(\frac{b}{a}{)}^r(\frac{a}{c}{)}^q\Rightarrow$

$\left(\frac{A{R}^{r-1}}{A{R}^{q-1}}{)}^p\right(\frac{A{R}^{q-1}}{A{R}^{p-1}}{)}^r(\frac{A{R}^{p-1}}{A{R}^{r-1}}{)}^q$

$\Rightarrow \left(R^{r-q}{)}^p\right(R^{q-p}{)}^r(R^{p-r}{)}^q$

$\Rightarrow R^{pr-pq+qr-pr+pq-rq}$

$\Rightarrow R^0=1$ option A