Question

Find the sum to n terms of the series $1\times 2\times 4+2\times 37+3\times 4\times 10+$

Answer 1

$Sumtonterms1\times 2\times 4+2\times 37+3\times 4\times 10+......$

$Gernalterms,ur=r(r+1)(r+3)=r(r^2+4r+3)$

$4r=r^3+4r^2+3r$

$\therefore Sn=\sum _{r=1}^{n}4r=\sum _{r=1}^n{r}^3+4\sum _{r=1}^n{r}^2+3\sum _{r=1}^{n}r$

$=\left[\frac{n(n+1)}{2}\right]+\frac{4}{6}n(n+1)(2n+1)+\frac{3}{2}n(n+1)$

$=\frac{n(n+1)}{12}\left[3n\right(n+1)+8(2n+1)+18]$

$=\frac{n(n+1)}{12}[3n^2+19n+26]$

$=\frac{n(n+1)}{12}[3n^2+13n+6n+26]$

$=\frac{n(n+1)}{12}\left[n\right(3n+13)+2(3n+13\left)\right]$

$=\frac{1}{12}n(n+1)(n+2)(3n+13)$

Hence this is the answer.