Question

If the $p^{th},q^{th}$ and $r^{th}$ terms of a G.P. are a, b and c respectively. Prove that $a^{q-r}{b}^{r-p}{c}^{p-q}=1$.

Answer 1

Let $A$ be the first term and $R$ be the common ratio. Then,

$a=pth$ term $=A{R}^{(p-1)}$

$b=qth$ term $=A{R}^{(q-1)}$

$c=rth$ term $=A{R}^{(r-1)}$

Substituting values of $a,b$ & $c$, we get,

$a^{(q-r)}.b^{(r-p)}.c^{(p-q)}$

$=[A{R}^{(p-1)}{]}^{(q-r)}.[A{R}^{(q-1)}{]}^{(r-p)}.[A{R}^{(r-1)}{]}^{(p-q)}$

$=A^{(q-r)}{R}^{(p-1)(q-r)}.A^{(r-p)}{R}^{(q-1)(r-p)}.A^{(p-q)}{R}^{(r-1)(p-q)}$

$=A^0{R}^0=1$