QJEEM24Ma7198: JEE Main 2024 Mathematics - Class 11 JEEM24Ma7198 Solution: Let the first three terms 2, p and q, with , of a G.P. be…

$\begin{aligned}& \text { Let } p=2 r, q=2 r^2 \\& T_7=2, T_8=2 r, T_{13}=2 r^2 \\& d=2 r-2=2(r-1) \\& 2 r^2=T_7+6 d=2+6(2)(r-1)=12 r-10 \\& \Rightarrow r^2-6 r+5=0 \\& \Rightarrow(r-1)(r-5)=0 \\& \therefore r=1,5 \\& r=1 \text { (rejected) as } q \neq 2 \\& \therefore r=5\end{aligned}$

$5^{\text {th }}$ term of G.P $=2 . r^4=2.5^4$

Let $1^{\text {st }}$ term of A.P $b a=a, d=8$

$2=a+(6)(8) \Rightarrow a=-46$

$\mathrm{n}^{\text {th }}$ term of A.P $=-46+(n-1) 8=8 n-54$

$\begin{aligned}& 2.5^4=8 n-54 \\& \Rightarrow 1250+54=8 n \\& \Rightarrow n=\frac{1304}{8}=163\end{aligned}$