Sequences and S...

The number of real roots of the equation $$1+a_{1}x+a_{2}x^{2}+....+a_{n}x^{n}=0$$ where $$|x| < \dfrac {1}{3}$$ and $$|a_{n}| < 2$$, is

If $${\left( {20} \right)^{19}} + 2\left( {21} \right){\left( {20} \right)^{18}} + 3{\left( {21} \right)^2}{\left( {20} \right)^{17}} + ... + 20{\left( {21} \right)^{19}} = k{\left( {20} \right)^{19}}$$ 
then $$k$$ is equal to

If $$(1+x)^{n}=C_{0}+C_{1}x+C_{2}x^{2}+....+...C_{n}x^{n}$$, then $$\dfrac { { C }_{ 1 } }{ { C }_{ 0 } } +\dfrac { { 2C }_{ 2 } }{ { C }_{ 1 } } +\dfrac { { 3C }_{ 3 } }{ { C }_{ 2 } } +....+\dfrac { { nC }_{ n } }{ { C }_{ n-1 } } =$$

The sum of the series $$^{20}C_{0}-^{20}C_{1}+^{20}C_{2}-^{20}C_{3}+-..+^{20}C_{10}$$ is$$\dfrac{1}{2}^{20}C_{10}$$

Let the $$n^{th}$$ terms of a series be given by
$$t_n$$ = $$\dfrac{n^2 -n -2}{n^2 + 3n}$$ , n 3 .
The product $$t_3$$, $$t_4$$,.........$$t_{50}$$ equals-

The sum of the series 11.2−12.3+13.411.2−12.3+13.4   is equal to 

Match the statements in List 1 with those in List 2

$$1-\dfrac { 3 }{ 2 } +\dfrac { 5 }{ 4 } -\dfrac { 7 }{ 8 } +...\dfrac { 3 }{ 2 } +\dfrac { 5 }{ 4 } -\dfrac { 7 }{ 8 } +...$$

If $${\left( {1 + x} \right)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ......... + {C_n}{x^n},n \in N$$. Then find the value of $$\displaystyle C_0^2 + \frac{{C_1^2}}{2} + {2^3}\frac{{{C_2}}}{3} + {2^4}\frac{{{C_3}}}{4} + ...... + {2^{n  }}\frac{{{C_n}}}{{n + 1}} $$

The value of $$\dfrac { ^{ a }{ C }_{ 9 } }{ n } +\dfrac { { ^{ n }C_{ 1 } } }{ n+1 } +\dfrac { ^{ n }{ C }_{ 2 } }{ n+2 } +.....+\dfrac { ^{ n }{ C }_{ H } }{ 2n } $$ is equal