Let the first three terms 2 , p and q , with q ? 2 , of a G.P. be respectively the 7th, 8th and 13th terms of an A.P. If the 5th term of the G.P. is the nth term of the A.P., then n is equal to [2024]

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Solution

Q.
Let the first three terms $2, p$ and $q,$ with $q \neq 2,$ of a G.P. be respectively the 7th, 8th and 13th terms of an A.P. If the 5th term of the G.P. is the $n^{th}$ term of the A.P., then $n$ is equal to [2024]

1 177

2 151

3 169

4 163

Ans.
(4)

Since, $2, p$ and $q$ are in G.P.

$∴$ $p^{2} = 2 q$ ...(i)

Let first term of the A.P. be $a$ and common difference be $d .$

$∴$    $T_{7} = a + 6 d = 2$ ...(ii)

$T_{8} = a + 7 d = p$ ...(iii)

and $T_{13} = a + 12 d = q$ ...(iv)

$∴$   From (ii) and (iii), we get $d = p - 2$

From (ii) and (iv), we get $6 d = q - 2 \Rightarrow 6 (p - 2) = q - 2$

$\Rightarrow 6 p = q + 10$ ...(v)

From (i) and (v), we get $p = 2$ or $p = 10$ $∴$    $q = 2$ or 50

But $q \neq 2$

Hence, $p = 10, q = 50$    $∴$ $d = 8$ and $a = - 46$

Since, 5th term of G.P. $= n^{th}$ term of A.P.

$∴$ $2 {(\frac{10}{2})}^{4} = - 46 + (n - 1) 8$

$\Rightarrow 1250 = - 46 + 8 n - 8 \Rightarrow 8 n = 1304 \Rightarrow n = 163$

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