Question

Find the sum of n terms of the series $1.3+4.7+9.13+16.21+25.31+....$

Answer 1

$\Rightarrow 14916\mathrm{25....}$

$357\mathrm{9....}$

$22\mathrm{2....}$

$u_n=n^2$

$371321\mathrm{31....}$

$468\mathrm{10....}$

$22\mathrm{2....}$

$u_n=3+4(n-1)(n-2)$

$=n^2+n+1$

$\therefore t_n$ (nth term)$=n^2(n^2+n+1)$

$=(n-1)(n+1)+1n(n+2)-n+1)$

$=(n-1)n(n+1)(n+2)+n(n+2)-(n-1)n(n+1)+(n-1)(n+1)+1-n$

$S_n=\sum t_n$

$t_n=(n-1)n(n+1)(n+2)+(n-1)n(n+1)+n(2n+1)$;

$\therefore S_n=\frac{1}{5}(n-1)n(n+1)(n+2)(n+3)-\frac{1}{4}(n-1)n(n+1)(n+2)+\frac{2}{3}n(n+1)(n+2)-\frac{1}{2}n(n+1)$

$S_n=\frac{n(n+1)\{12+33n^2+31n+8\}}{60}$