Sequences and S...

The sum of the series $$1^2 - 2^2 + 3^2 - 4^2 + .......... - 1000^2 + 1001^2$$ is

 $$ 1.4+2.7+3.10+...+\mathrm{n}(3\mathrm{n}+1)=$$

The sum of $$7$$ terms of the series $$1^2-2^2+3^2-4^2+5^2-6^2+...$$ is 

 $$1^{2}+1+2^{2}+2+3^{2}+3+\ldots..+n^{2}+n=$$

$$\displaystyle \frac{1.2^{2}+2.3^{2}+3.4^{2}+.\cdot.\cdot.\cdot.\cdot+n(n+1)^{2}}{1^{2}.2+2^{2}.3+3^{2}.4++n^{2}(n+1)}=$$

$$1^3+1^2+1+2^{3}+2^{2}+2+3^{3}+3^{2}+3+ ...  3\mathrm{n} $$ terms $$=$$ 

The value of $$\sum_{r=1}^{10}\left ( 2^{r-1}+8r-3 \right )$$ is equal to

$$\displaystyle \frac{1^{2}}{1}+\frac{1^{2}+2^{2}}{1+2}+\frac{1^{2}+2^{2}+3^{2}}{1+2+3}+\ldots+n$$ terms $$=$$

Find $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{(2n-1)!}$$

Evaluate $$1.6+2.9+3.12+...+\mathrm{n}(3\mathrm{n}+3)=$$