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23.   Find the sum of the first n terms of the series: 3+ 7 +13 +21 +31 +…

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The series: 3+ 7 +13 +21 +31 +…..............

n th term  = n^2+n+1=a_n

S_n=\sum _{k=1}^{n} a_k=\sum _{k=1}^{n} k^{2}+k+1

                        =\sum _{k=1}^{n} k^2+\sum _{k=1}^{n} k+\sum _{k=1}^{n} 1

                     =\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}+n

                   =n\left ( \frac{(n+1)(2n+1)}{6}+\frac{n+1}{2}+1 \right )

                   =n\left ( \frac{(n+1)(2n+1)+3(n+1)+6}{6} \right )

                  =n\left ( \frac{2n^2+n+2n+1+3n+3+6}{6} \right )

                  =n\left ( \frac{2n^2+6n+10}{6} \right )

                   =n\left ( \frac{n^2+3n+5}{3} \right )

Posted by

seema garhwal

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