# Find the sum of $n$ terms of the series $3+7+13+21+31+.............$

$\begin{array}{1 1} \large\frac{n}{3}\big[n^2+3n+5\big] \\ \large\frac{n}{3}\big[n^2-3n+5\big] \\ \large\frac{n}{3}\big[n^2+3n-5\big] \\ \large\frac{n}{3}\big[n^2+4n+5\big]\end{array}$

Toolbox:
• Sum of $n$ terms of ant series is given by $S_n=\sum t_n$
• um of $n-1$ terms of an A.P. $=\large\frac{n-1}{2}$$[2a+(n-2)d] • \sum n^2=\large\frac{n(n+1)(2n+1)}{6} • \sum n=\large\frac{n(n+1)}{2} • \sum 1=n Given series is is written as S_n=3+7+13+21+31+.................+t_n+0..........(i) S_n=0+3+7+13+21+31+.................+t_n.........(ii) Subtracting (i) - (ii) we get 0=3+(4+6+8+10+............(n-1)\: terms)-t_n \Rightarrow\:t_n=3+(4+6+8+10+.........(n-1)\:terms) 4+6+8+..........(n-1) \:terms is an A.P. with 1^{st} term =a=4 common difference =d=2 and no. of terms =n-1 We know that sum of n-1 terms of an A.P. =\large\frac{n-1}{2}$$[2a+(n-2)d]$
$\therefore 4+6+8+......=\large\frac{n-1}{2}$$\big [2\times 4+(n-1-1)2\big] \qquad\:=\large \frac{n-1}{2}$$[8+(n-2)2]=(n-1)(2+n)=n^2+n-2$
Substituting this value in $t_n$ we get
$\Rightarrow\:t_n=3+(n^2+n-2)$
$i.e.,$ $t_n=n^2+n+1$
Step 2
Now for any series Sum of $n$ terms $=S_n=\sum t_n$
$\Rightarrow\:S_n=\sum (n^2+n+1)$
$\qquad =\sum n^2+\sum n+\sum 1$
$\qquad \:=\large\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}+$$n \qquad\:=n\bigg[\large\frac{(n+1)(2n+1)}{6}+\large\frac{n+1}{2}+$$1\bigg]$
$\Rightarrow\:S_n=\large\frac{n}{6}$$\big[2n^2+3n+1+3n+3+6\big]. \Rightarrow\:S_n=\large\frac{n}{6}$$\big[2n^2+6n+10\big]=\large\frac{n}{3}$$\big[n^2+3n+5\big]$