Sequences and S...

The series $$\dfrac{8}{5}+\dfrac{16}{65}+\dfrac{24}{325}+.........$$ upto infinity has the sum equal to 

How many Pythagorean triangles are there with integer sides and one leg is Ramanujan Number 1729 .

Find the missing number, if a certain rule is followed

$$\sum\limits_{r = 1}^n {{{\sin }^{ - 1}}\left( {\frac{{\sqrt r  - \sqrt {r - 1} }}{{\sqrt {r\left( {r + 1} \right)} }}} \right)} $$ is equal to 

Let $$A$$ be the sum of the first $$20$$ terms and $$B$$ be the sum of the first $$40$$ terms of the series $$1$$ + $$2.2^2$$ + $$3^2$$ + $$2.4^2$$ + $$5^2+2.6^2$$ +......... Find the value of $$A$$.

If $$1^{2}+2^{2}+3^{2}++(2003) ^{2}=(2003)(4007)(334)$$ and $$(1)(2003)+ (2)(2002)= (2003)(334)(x)$$ then $$x$$ equals

If $$\left( { 1-y } \right) ^{ 35 }\left( 1+y \right) ^{ 45 }={ A }_{ 0 }+{ A }_{ 1 }Y+{ A }_{ 2 }{ Y }^{ 2 }+{ A }_{ 3 }{ Y }^{ 3 }+.....+{ A }_{ 80 }{ Y }^{ 80 },$$ then

If $$n\in N$$ then $$\displaystyle \sum \limits_{ r=0 }^{ n }{ { \left( -1 \right)^r  }\;^{ n }{ C }_{ 2 } } \left[ \dfrac { 1 }{ { 2 }^{ r} }+ \dfrac { { 3 }^{ r } }{ { 2 }^{ 2r } }+ \dfrac { { 7 }^{ r } }{ { 2 }^{ 3r } } +\;.\;.\;.\;\;upto\quad m\quad terms \right] $$ is equal to

If $$ '-'$$ denotes $$'\div ',\ '\div '$$ denotes $$'\times ',\ '+'$$ denotes' - 'and $$'\times '$$ denotes $$'+'$$, then find the value of $$116+9\div 52-4\times 5$$.

$$1+\dfrac{1}{2}(1+2)+\dfrac{1}{3}(1+2+3)+\dfrac{1}{4}(1+2+3+4)+.....$$ upto $$20$$ terms is