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# If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

Q : 9    If the sum of first $\small 7$ terms of an AP is $\small 49$ and that of $\small 17$ terms is $\small 289$ , find the sum of
first $\small n$ terms.

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It is given that
$S_7 = 49 \ and \ S_{17}= 289$
Now, we know that
$S_n = \frac{n}{2}\left \{ 2a+(n-1)d \right \}$
$\Rightarrow S_{7}= \frac{7}{2}\left \{ 2\times(a) +(7-1)d\right \}$
$\Rightarrow 98= 7\left \{ 2a +6d\right \}$
$\Rightarrow a +3d = 7 \ \ \ \ \ \ \ -(i)$
Similarly,
$\Rightarrow S_{17}= \frac{17}{2}\left \{ 2\times(a) +(17-1)d\right \}$
$\Rightarrow 578= 17\left \{ 2a +16d\right \}$
$\Rightarrow a +8d = 17 \ \ \ \ \ \ \ -(ii)$
On solving equation (i) and (ii) we will get
a = 1 and d = 2
Now, the sum of first n terms is
$S_n = \frac{n}{2}\left \{ 2\times 1 +(n-1)2 \right \}$
$S_n = \frac{n}{2}\left \{ 2 +2n-2 \right \}$
$S_n = n^2$
Therefore, the sum of n terms  is  $n^2$

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