Sequences and S...

The sum to the first n terms of the series  $$\dfrac{1}{2} + \dfrac{3}{4} + \dfrac{7}{8} + \dfrac{{15}}{{16}} + {\rm{ }} \ldots $$ is $$9 + {2^{ - 10}}$$.  The value of n is

$$A = (2 + 1)({2^2} + 1)({2^4} + 1).....\left( {{2^{2016}} + 1} \right)$$. The value of $${\left( {A + 1} \right)^{\frac{1}{{2016}}}}$$ is

The value of $$S=\sum _{ k=1 }^{ 6 }{ \left( \sin { \dfrac { 2\pi k }{ 7 }  } -i\cos { \dfrac { 2\pi k }{ 7 }  }  \right)  }$$ ?

If $$P,\,Q$$ be two arithmetic means between $$\dfrac{1}{3}$$ and $$\dfrac{1}{24}$$, then their value are 

If $$\dfrac{1}{x (x + 1) (x + 2)....(x + n)} = \dfrac{A_0}{x} + \dfrac{A_1}{x + 1} + \dfrac{A_2}{x + 2} + .... + \dfrac{A_n}{x + n} $$, then $$ A_r$$ $$=$$

What is the sum of the series $$\dfrac{1}{{{3^2} - 4}} + \dfrac{1}{{{7^2} - 4}} + \dfrac{1}{{{{11}^2} - 4}} + ..... + \dfrac{1}{{{{39}^2} - 4}}$$

Number of identical terms in the sequence $$2, 5, 8, 11,$$___ upto $$100$$ terms and $$3, 5, 7, 9, 11$$____ upto $$100$$ terms are

Find the value of ?. 

The value of the expression $$\left(\dfrac{1}{2^2 -1}\right) + \left(\dfrac{1}{4^2 -1}\right) + \left(\dfrac{1}{6^2 -1}\right) + ... + \left(\dfrac{1}{20^2 -1}\right)$$ is

$$67,84,95,.,133,158$$