Let the first three terms $2, p$ and $q$, with $q
eq 2$, of a G.P. be respectively the $7^{ ext {th }}, 8^{ ext {th }}$ and $13^{ ext {th }}$ terms of an A.P. If the $5^{ ext {th }}$ term of the G.P. is the $n^{ ext {th }}$ term of the A.P., then $n$ is equal to:

Question

Let the first three terms $2, p$ and $q$, with $q 
eq 2$, of a G.P. be respectively the $7^{	ext {th }}, 8^{	ext {th }}$ and $13^{	ext {th }}$ terms of an A.P. If the $5^{	ext {th }}$ term of the G.P. is the $n^{	ext {th }}$ term of the A.P., then $n$ is equal to:

Aicharya · Accepted Answer

$p^2=2 q$ $2=a+6 d$ ...(i) $\mathrm{p}=\mathrm{a}+7 \mathrm{~d}$ ...(ii) $q=a+12 d$ ...(iii) $\mathrm{p}-2=\mathrm{d}$ ((ii) - (i)) $q-p=5 d$ ((iii) - (ii)) $q-p=5 d$ $\begin{aligned} & q-p=5(p-2) \ & q=6 p-10 \ & p^2=2(6 p-10) \ & p^2-12 p+20=0 \ & p=10,2 \ & p=10 ; q=50 \ & d=8 \ & a=-46 \ & 2,10,50,250,1250 \ & a^4=a+(n-1) d \ & 1250=-46+(n-1) 8 \ & n=163\end{aligned}$

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7^{\text {th }},…

Solution