Sequences and S...

The sum to infinity of the series:
$$\dfrac {3}{{1}^{3}}+\dfrac {5}{{1}^{3}+{2}^{3}}+\dfrac {7}{{1}^{3}+{2}^{3}+{3}^{3}}+..$$ is-

Sum to $$n$$ terms of the series$$\dfrac { 1 }{ 1.2.3.4 } +\dfrac { 1 }{ 2.3.4.5 } +\dfrac { 1 }{ 3.4.5.6 } +..........$$, is

If $$\dfrac{x}{0.2} + \dfrac{x}{0.3} + \dfrac{x}{0.6} + \dfrac{x}{0.4} + \dfrac{x}{0.5} = 87$$, then the value of x is equal to 

The value of $$\displaystyle\sum^{100}_{r=2}\dfrac{3^r(2-2r)}{(r+1)(r+2)}$$ is equal to?

If $$f(x)=\displaystyle\Pi^3_{i=1}(x-a_i)+\displaystyle\sum^3_{i=1}a_i-3x$$ where $$a_i < a_{i+1}$$ for $$i=1, 2,$$ then $$f(x)=0$$ has:

The sum of the series $$1+\dfrac{1}{1!}.\dfrac{1}{4}+\dfrac{1\cdot 3}{2!}\left(\dfrac{1}{4}\right)^{2}+\dfrac{1\cdot 3 \cdot 5}{3!} \left(\dfrac{1}{4}\right)^{3}+........$$ to $$\infty$$ is ?

For natural numbers m, n if $${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n} = 1 + {a_1}y + {a_2}{y^2} + .......$$ and $$a_1 = a_2 = 10$$, Then $$(m,n)$$ is

The value of $$(21{C}_{1}-10{C}_{1})+(21{C}_{2}-10{C}_{2})+(3{C}_{1}-10{C}^{3})+(21{C}_{4}-10{C}^{4})+.....(21{C}_{10}-10{C}_{10})$$ is