# sum of series $(2^2-1)(6^2-1)+(4^2-1)(8^2-1)+\cdots \cdots +\cdots (100^2-1)(104^2-1)$

The sum of series $(2^2-1)(6^2-1)+(4^2-1)(8^2-1)+\cdots \cdots +\cdots (100^2-1)(104^2-1)$

Attempt Assume $\displaystyle S = \sum^{50}_{r=1}((2r)^2-1)((2r+4)^2-1) = \sum^{50}_{r=1}(4r^2-1)(4r^2+16r+15)$

$\displaystyle S = \sum^{50}_{r=1}(16r^4+64r^3+56r^2-16r-15)$

could some help me how to solve it, thanks

• The sums $\sum_{r = 1}^{50}r, \sum_{r = 1}^{50}r^2, \sum_{r = 1}^{50}r^3$ and $\sum_{r = 1}^{50}r^4$ may be solved separately. – Arthur Jan 16 '17 at 6:55
• @Arther how i solve $\sum^{50}_{r=1}r^4,$ please explain me – DXT Jan 16 '17 at 6:56
• I googled and found this, which looks like it might be helpful. – Arthur Jan 16 '17 at 6:58

Since $$((2r)^2-1)((2r+4)^2-1)=(2r-1)(2r+1)(2r+3)(2r+5)$$ we get \small\begin{align}&\sum_{r=1}^{50}(2r-1)(2r+1)(2r+3)(2r+5)\\\\&=\frac{1}{10}\sum_{r=1}^{50}\color{red}{10}(2r-1)(2r+1)(2r+3)(2r+5)\\\\&=\frac{1}{10}\sum_{r=1}^{50}\color{red}{((2r+7)-(2r-3))}(2r-1)(2r+1)(2r+3)(2r+5)\\\\&=\frac{1}{10}\sum_{r=1}^{50}((2r+7)(2r+5)(2r+3)(2r+1)(2r-1)-(2r+5)(2r+3)(2r+1)(2r-1)(2r-3))\\\\&=\frac{1}{10}(107\cdot 105\cdot 103\cdot 101\cdot 99-7\cdot 5\cdot 3\cdot 1\cdot (-1))\end{align}
$$S_n = \sum^{n}_{r=1}(16r^4+64r^3+56r^2-16r-15)=\frac{1}{5} \left(16 n^5+120 n^4+280 n^3+180 n^2-71 n\right)$$