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Q.
The fourth, seventh and tenth terms of a G.P. are $p, q, r$ respectively, then:
Sequences and Series
Solution:
Let $a$ be the first term and $r$ be common ratio.
Fourth term of G.P. $: p=T_{4}=a r^{3}\, \dots(i)$
Seventh term of GP. : $q=\frac{4}{T_{7}}=a r^{6}....(ii) $
Tenth term of G.P. : $r=T_{10}=a r^{9}\, \dots(iii)$
Equ. (i) $\times$ Equ. (iii):
$p r=a r^{3} \times a r^{g} $
$\Rightarrow p r=a^{2} r^{12} $
$\Rightarrow p r=\left(a r^{6}\right)^{2}$
$ \Rightarrow p r=q^{2}$