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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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