Sequences and S...

If $${ a }_{ n }=\sum _{ r=0 }^{ n }{ \cfrac { 1 }{ { _{  }^{ n }{ C } }_{ r } }  } $$, the value of $$\sum _{ r=0 }^{ n }{ \cfrac { n-2r }{ { _{  }^{ n }{ C } }_{ r } }  } $$

Observe the pattern carefully
$$11\times11=121$$
$$111\times111=12321$$
$$1111\times1111=\,?$$

The sum of the series 
$$(2)^2+2(4)^{2}+3(6)^{2}+....$$ upto $$10$$ terms is

The sum of the series $$1+\frac { 1.3 }{ 6 } +\frac { 1.3.5 }{ 6.8 } +....\infty$$ is 

If $$A_k=\begin{bmatrix} k & k-1\\ k-1 & k\end{bmatrix}$$ then $$|A_1|+|A_2|+..+|A_{2015}|=?$$

The value of $$\sum^{10}_{x=1}\sum^{r=x-1}_{r=0}(2^{x}-2^{r})$$ is

Let $$a=\dfrac{1^{2}}{1}+\dfrac{2^{2}}{3}+\dfrac{3^{2}}{5}+......+\dfrac{(1001)^{2}}{2001}$$, $$b=\dfrac{1^{2}}{3}+\dfrac{2^{2}}{5}+\dfrac{3^{2}}{7}+......+\dfrac{(1001)^{2}}{2003}$$. The closest integer of $$a-b$$ is

The value of $$1 + \dfrac{x \, \log_e \, 2}{1!} + \dfrac{x^2}{2!} (\log_e 2)^2 + \dfrac{x^3}{3!} (\log_e 2)^3+ ... \infty$$s equal to 

If $$z=\dfrac { 1 }{ 3 } +\dfrac { 1.3 }{ 3.6 } +\dfrac { 1.3.5 }{ 3.6.9 } +..........,\infty $$ then 

Sum of the series 
$$(1\times 2015)+(2 \times 2014)+(3\times 2013)......+(2015\times 1)$$ is equal to-