Sequences and S...

In a triangle $$ABC$$, 
$$acos^2(\frac{C}{2})+c\;cos^2(\frac{A}{2})=\dfrac{3b}{2}$$, then the sides $$a,b,c$$ 

Let $$N=\{1,2,3,.......,10\},$$ then sum of product of number taken two at a time from the set $$N$$ is 

The sum $$\sum _{ i=0 }^{ m }{ \left( \begin{matrix} 10 \\ i \end{matrix} \right) \left( \begin{matrix} 20 \\ m-i \end{matrix} \right)  }$$ (where $$\left( \begin{matrix} p \\ q \end{matrix} \right)=0$$ if $$p<q$$) is maximum where $$m$$ is

$$1.3.4+2.5.8+3.6.9+$$ upto $$n$$ terms is equal to ________ .

If $$64a^2+36b^2=400, ab=4 $$ , then $$8a+6b$$ is: 

The sum of the series  
$$1+2.2+3.2^{2}+4.2^{3}+5.2^{4}+...+1000.2^{999}$$ is

The sum of the series $$10.^{n}C_{0}+10^{2}.^{n}C_{1}+10^{3}.^{n}C_{2}+...10^{n+1}.^{n}C_{n}$$ is 

If $$| x| < 1$$ , then the sum of series $$ 1+ 2x + 3x^2 + 4x^3 + .........\infty$$ will be

There is a defined relationship between the pair of figure on either side of : :. Identify the relationship of the given pair and find be the missing figure.

$$\displaystyle\sum^n_{r=1}\displaystyle\sum^{r-1}_{p=0}$$ $$^{n}C_r\cdot {^{r}C_p}\cdot 2^p$$ is equal to?