Sequences and S...

Sum to n terms the following series :

$$\sum\limits_{r = 0}^{10} {r{.^{10}}{C_r}{{.3}^r}{{\left( { - 2} \right)}^{10 - r}}} $$ is equal 

The value of the expression $$\sum _{ r=0 }^{ n }{ { (-1) }^{ r } } \left( \dfrac { ^nC_r  }{^{r+3}C_r  }  \right) $$ is

$$S=\tan^{-1}\left(\dfrac{1}{n^2+n+1}\right)+\tan^{-1}\left(\dfrac{1}{n^2+3n+3}\right)+.....+\tan^{-1}\left(\dfrac{1}{1+(n+19)(n+20)}\right)$$, then $$\tan S$$ is equal to?

Evaluate:-
If $$\sum\limits_{r - 0}^n {{{\left\{ {\frac{{^n{C_{r - 1}}}}{{^n{C_r}{ + ^n}{C_{r - 1}}}}} \right\}}^3} = \frac{{25}}{{24}}} $$

Sum of the series
$$S=1^{2}-2^{2}+3^{2}-4^{2}+..... -2000^{2}+2003^{2}$$ is

If the expansion of $$\left( x+a \right) ^{ n }$$ if the sum of odd terms be P & sum of even terms be Q, prove that

If $$\left| x \right| <$$ 1 and $$\left| y \right| <$$ 1, then the sum of infinity of the series $$(x+y)+(x^2+xy+y^2)+(x^3+x^2y+xy^2+y^3)+.....$$ to $$\infty $$ is 

$$\sum _{ r=1 }^{ n }{ r. } $$ $$^{2n}C_r$$ is equal to