Sequences and S...

The $$(n+1)^{th} $$ term from the end in $$(x - \frac{1}{x})^{3n}$$ is 

The sume of the series $$1^3 - 2^3 + 3^3 - ........ + 9^3$$ =

If $$a_{1}, a_{2}, ......., a_n(n > 3)$$ are all unequal positive real numbers, and 

$$E = \dfrac{(1 + a_{1} + a_{1}^{2})(1 + a_{2} + a_{2}^{2})......(1 + a_{n} + a_{n}^{2})}{a_{1}, a_{2}, ......., a_{n}}$$
 then which of the following best describes E?

The sum of the series $${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + .......$$ up to low upon to n term is to the 

The positive integer n for which $$2 \times {2^2} + 3 \times {2^3} + 4 \times {2^4} + ....... + n \times {2^n} = {2^{^{n + 10}}}$$ is______

The sum of infinite series $$\dfrac{1.3}{2}+\dfrac{3.5}{2^2}+\dfrac{5.7}{2^3}+\dfrac{7.9}{2^4}+...\infty$$.

The sum to infinite of the series
$$1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + ........$$

If $$ \displaystyle \lim _{ x\rightarrow 0^+ }{ x\left( \left[ \dfrac { 1 }{ x }  \right] +\left[ \dfrac { 5 }{ x }  \right] +\left[ \dfrac { 11 }{ x }  \right] +\left[ \dfrac { 19 }{ x }  \right] +\left[ \dfrac { 29 }{ x }  \right] +.......to\quad n\quad terms \right)  }=430$$ (where [.] denotes the greatest integer function), then $$n=$$

If $$\displaystyle\sum _{ n=1 }^{ 2013 }{ \tan { \left( \dfrac { \theta  }{ { 2 }^{ n } }  \right)  }  } \sec { \left( \dfrac { \theta  }{ { 2 }^{ n-1 } }  \right)  } =\left( \dfrac { \theta  }{ { 2 }^{ a } }  \right) -\left( \dfrac { \theta  }{ { 2 }^{ b } }  \right)$$ then $$(b+a)$$ equals 

If $$\sum^5_{n=1}\dfrac{1}{n(n+1)(n+2)(n+3)}=\dfrac{k}{3}$$, then k is equal to?