Sequences and S...

$$\displaystyle \sum_{r = 0}^{n}{ \left( \frac{ 2^{r-2} . ^nC_r}{(r+1)(r+2)} \right) }$$ is equal to

If the $$p^{th}$$ term of the series of positive numbers $$25, 22\dfrac {3}{5}, 20\dfrac {1}{2}, 18\dfrac {1}{4}$$, .... is numerically the smallest, then the $$p^{th}$$ is.

13, 74, 290, 650,.......

If $$\begin{vmatrix} x \end{vmatrix}<1$$  then the coefficient of $$x^5$$ in the expansion of $$\dfrac{3x}{(x-2) (x-1)}$$ is

If the coefficients of $$x^9,x^{10},x^{11}$$ in the expansion of $$(1+x)^n $$ are in arithmetic progression then $$n^2-41n=$$

If $$\sin^{-1}\begin{pmatrix}  x-\frac{x^2}{2}+\frac{x^3}{4}-..........\infty  \end{pmatrix}+\cos^{-1}\begin{pmatrix} x^2-\frac{x^4}{2}+\frac{x^6}{4}........\infty\end{pmatrix} =\frac{\pi}{2}$$ and $$0<x<\sqrt{2}$$ then x=

The value of $$\cfrac { 1 }{ \left( 2n-1 \right) !0! } +\cfrac { 1 }{ \left( 2n-3 \right) !2! } +\cfrac { 1 }{ \left( 2n-5 \right) !4! } +....+\cfrac { 1 }{ 1!\left( 2n-2 \right) ! } $$ equal to

The sum of first $$20$$ terms of the series $$1,6,13,22$$- is

$$2.4+4.7+6.10+.....$$ upto $$(n-1)$$ terms 

Sum of the series $$\sum _{ r=1 }^{ n }{ \left( { r }^{ 2 }+1 \right) r! } $$ is ______