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

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

$5^{\text {th }}$ 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 $=-46+(n-1) 8=8 n-54$

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

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

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

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