Sequences and S...

For some natural $$N$$ , the number of positive integral $$x$$ satisfying the equation, 
$$1!+2!+3!+......+(x)!=(N)^2$$ is :

Arrange these numbers in ascending order. 
$$756, 567, 657, 676$$ 

$$If\quad { sin }^{ -1 }\left( x-\frac { { x }^{ 2 } }{ 2 } +\frac { { x }^{ 3 } }{ 4 } -......\infty  \right) +{ cos }^{ -1 }\left( { x }^{ 2 }-\frac { { x }^{ 4 } }{ 2 } +\frac { { x }^{ 6 } }{ 4 } -.....\infty  \right) =\frac { \pi  }{ 2 } \quad and\quad 0<x<\sqrt { 2 } \quad then\quad x=$$

If $$S_n=\sum\limits_{r=1}^n \dfrac{2r+1}{r^4+2r^3+r^2}$$ then $$S_{20}$$ =

The sum to $$50$$ terms of the series $$\dfrac {1}{2} + \dfrac {3}{4} + \dfrac {7}{8} + \dfrac {15}{16} + ....$$ is equal to

$$\sum\limits_{r=1}^{50}\Big[ \dfrac{1}{49+r}-\dfrac{1}{2r(2r-1)}\Big]=$$

The value of $$x$$ satisfying the equation $$\dfrac{5050-\left(\dfrac{1}{2}+\dfrac{2}{3}+\dfrac{3}{4}+......+\dfrac{5049}{5050}\right)}{1+\dfrac{1}{2}+\dfrac{1}{3}+......+\dfrac{1}{5050}}=\dfrac{x}{5050}$$ is 

If $$\left| a \right| < 1$$ and $$\left| b \right| < 1$$ then $$\eqalign{  &   \cr   & S = 1 + \left( {1 + a} \right)b + \left( {1 + a + {a^2} } \right){b^2} + ...} $$=

If $$\left| a \right| < 1$$ and $$\left| b \right| < 1$$ then $$s = 1 + \left( {1 + a} \right)b + \left( {1 + a + {a^2}} \right){b^2} + \left( {1 + a + {a^2} + {a^3}} \right){b^3} + ...is - .$$

If $$1 + {x^2} = \sqrt {3}x $$, then $$\displaystyle \prod\limits_{n = 1}^{24} {\left( {x^n} + \dfrac {1} {x^n} \right)} $$ is equal to