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nth Term of an AP

Question 25 I...

Question

Question 25
In an AP, if $S_{n} = 3 n^{2} + 5 n$ and $a_{k} = 164$ , then find the value of $k$ .

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Solution

nth term of an AP,
$a_{n} = S_{n} - S_{n - 1}$
$= 3 n^{2} + 5 n - 3 (n - 1)^{2} - 5 (n - 1) [∵ S_{n} = 3 n^{2} + 5 n (g i v e n)]$
$= 3 n^{2} + 5 n - 3 n^{2} - 3 + 6 n - 5 n + 5$
$a_{n} = 6 n + 2 \dots (i)$
$O r a_{k} = 6 k + 2 = 164 [∵ a_{k} = 164 (g i v e n)]$
$\Rightarrow 6 k = 164 - 2 = 162$
$∴ k = 27$

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