Question

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

Answer 1

Lrt A be the first term and R be the common ratio of given G.P

$\therefore a_p=a\Rightarrow A{R}^{p-1}=a......\left(1\right)$

$a_q=b\Rightarrow A{R}^{q-1}=b......\left(1\right)$

$a_r=c\Rightarrow A{R}^{r-1}=c......\left(1\right)$

Now, $a^{q-1},b^{r-p},c^{p-q}$

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

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

= $A^{q-r+r-p+p-q}.R^{pq-pr-q+r+qr-pq-r+p+pr-qr-p+q}$

= $A^{\circ }.R^{\circ }=1.1=1$

Thus, $a^{q-r}.b^{r-p}.c^{p-q}=1$