Find the sum of the first n terms of the series : `3 + 7 + 13 +21 + 31 +dot dot dot`

Find the sum of the first n terms of the series : `3 + 7 + 13 +21 + 31

1.	Find the sum of the first n terms of the series : `3 + 7 + 13 +21 + 31 +dot dot dot`
Answer» Let `S_(n) =3+7+13+21+31+...+T_(n-1)+T_(n)` `S_(n) =3+7+13+21+...+T_(n-2)+T_(n-1)+T_(n)` On subtraction, we get `0=3+[4+6+8+10+..."to"(n-1)" terms"]-T_(n)` `rArr T_(n)=3+[4+6+8+... "to"(n-1)"terms"]` `=3+((n-1))/(2)xx[2xx4+(n-2)xx2]` `=3+(n-1)(n+2)=(n^(2)+n+1).` `therefore S_(n)=sum_(k=1)^(n)T_(k) =sum_(k=1)^(n)(k^(2)+k+1)` `=sum_(k=1)^(n)k^(2)+sum_(k=1)^(n)k+n " "[because 1+1+1+... "to n terms"=n]` `=(1)/(6)n(n+1)(2n+1)+(1)/(2)n(n+1)+n=(1)/(3)n(n^(2)+3n+5).`