Sequences and S...

$$496 : 204 : : 329 : ?$$

If $$x_{1},x_{2},....$$ are in H.P and $$x_{1},2,x_{20}$$ are in G.P, then $$\displaystyle \sum _{ 1 }^{ 19 }{ x_{r} .x_{r+1}}=$$ 

If $$S_n=\dfrac{7}{4.1.2}+\dfrac{10}{4^2.2.3}+\dfrac{13}{4^3.3.4}+....$$ then $$S_{\propto}$$ is equal to?

If $${ sin }^{ 4 }\alpha .{ cos }^{ 2 }\alpha =\sum _{ r=0 }^{ 3 }{ { C }_{ r } } .cos(2r\alpha )$$ then $${ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 }+{ C }_{ 3 }=$$

The value of  $$2 ^ { n } \{ 1.3 .5 \ldots \ldots ( 2 n - 3 ) ( 2 n - 1 ) \}$$  is

The value of the series $$\dfrac{1}{3!}+\dfrac{2}{5!}+\dfrac{3}{7!}+..... $$  to $$\infty $$   is equal to 

The sum of the series
  $$\dfrac { 9 }{ 5^{ { 2 } }\cdot 2.1 } +\dfrac { 13 }{ 5^{ { 3 } }\cdot 3.2 } +\dfrac { 17 }{ 5^{ { 4 } }\cdot 4.3 } ..........$$  Infinite terms

The sum to infinity of the series $$1+\dfrac {2}{3}+\dfrac { 6 }{ { 3 }^{ 2 } } +\dfrac { 10 }{ { 3 }^{ 3 } } +\dfrac { 14 }{ { 3 }^{ 4 } }+.... $$ is

The sum to  $$50$$  terms of the series  $$\dfrac { 3 } { 1 ^ { 2 } } + \dfrac { 5 } { 1 ^ { 2 } + 2 ^ { 2 } } + \dfrac { 7 } { 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } } + \ldots$$  is

If $$an =\displaystyle \sum_{r =0}^{n} \dfrac{1}{^{n}C_{r}}$$ then $$\displaystyle \sum_{r=0}^{n} \dfrac{r}{^{n}C_{r}}$$ equals