Question

If $a,b,c$ are $p^{th}$, $q^{th}$, and $m^{th}$ terms of a G.P, then the value of $\left(\frac{c}{b}{)}^p\right(\frac{b}{a}{)}^m(\frac{a}{c}{)}^q$ is equal to ____.

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

Answer 1

The correct option is C 1

Let 'A' be the initial Term, and 'R' be the common Ratio.

$a=A{R}^{p-1}$,
$b=A{R}^{q-1}$,
$c=A{R}^{m-1}$
$[\because T_n=a{r}^{n-1}]$

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

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

$=R^{(m-q)p+(q-p)m+(p-m)q}$

$=R^{mp-qp+qm-pm+pq-qm}$

$=R^0$

$=1$