The sum 31^31^2+5(1^3+2^3)1^2+2^2+7(1^3+2^3+3^3)1^2+2^2+3^2+ up to the 10th term is:

Question

The sum $\dfrac{3	imes1^3}{1^2}+\dfrac{5(1^3+2^3)}{1^2+2^2}+\dfrac{7(1^3+2^3+3^3)}{1^2+2^2+3^2}+\cdots$ up to the 10th term is:

Accepted Answer

$$T_r = \frac{(2r+1)\left(\frac{r(r+1)}{2}\right)^2}{\frac{r(r+1)(2r+1)}{6}} = \frac{3r(r+1)}{2}.$$

$$S_{10} = \frac32\sum_{r=1}^{10}(r^2+r) = \frac32\left[\frac{10\times11\times21}{6}+\frac{10\times11}{2}\right] = \frac32\times5\times11\times8 = 660.$$

Solutions · 1