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Sum of n Terms

The sum of fi...

Question

The sum of first $n$ terms of an AP is $3 n^{2} - n$ . Find the $25^{t h}$ term of this AP.

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Solution

Given $S_{n} = 3 n^{2} - n$

Then $S_{n - 1} = 3 {[3 (n - 1)]}^{2} - (n - 1)$

$a_{n} = S_{n} - S_{n - 1}$

$\Rightarrow a_{n} = [3 n^{2} - n] - [3 (n - 1)^{2} - (n - 1)]$

$= 3 [n^{2} - (n - 1)^{2}] - n + n - 1$

$= 3 (n^{2} - n^{2} + 2 n - 1) - 1 = 3 (2 n - 1) - 1 = 6 n - 4$

Then, $a_{25} = 6 \times 25 - 4 = 150 - 4 = 146$

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