Sequences and S...

If j, k, and n are consecutive integers such that $$0 < j < k < n$$ and the units (ones) digit of the product jn is 9, what is the units digit of k ? 

The sum of the series $$\sum _{ n=1 }^{ \infty  }{ \sin { \left( \cfrac { n!\pi  }{ 720 }  \right)  }  } $$ is

$$\displaystyle\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\cdots\cdots+\frac{1}{n(n+1)}=$$

The variance of the series
$$a,a+d,a+2d,.....a+(2n-1)d,a+2nd$$ is

It is given that $$\sum_{r = 1}^{\infty} \dfrac{1}{(2 r - 1)^2} = \dfrac{\pi^2}{8}$$, then $$\sum_{r = 1}^{\infty} \dfrac{1}{r^2}$$ is equal to

Let $$\{a_n\}$$ be a sequence of numbers satisfying the relation $$(3-a_{n+1})(6+a_n)=18$$ for all $$n\ge 0$$ and $$a_0=3$$. Then
$$\displaystyle \underset{n\rightarrow \infty}{lim}\dfrac{1}{2^{n+2}}\sum_{j=0}^{n}\dfrac{1}{a_j}$$

If $$\displaystyle \sum _{ r =1 }^{n}{ t_r = \frac{n(n+1)(n+2)(n+3)}{8} }$$, then $$\displaystyle \sum _{r = 1}^{n}{ \frac{1}{t_r}} $$ equals

Let $$A=16{-4}+2^{-4}+3^{-4}+4^{-4}+...$$ and $$B=1^{-4}+3^{-4}+5^{-4}+7^{-4}+...$$. The ratio $$\dfrac{A}{B}$$ in the lowest form is

Which of the following options will continue the given series?
$$29$$, $$34$$, $$32$$, $$37$$, $$35$$, ?

The value of $$S=\sqrt { 1+\cfrac { 1 }{ { 1 }^{ 2 } } +\cfrac { 1 }{ { 2 }^{ 2 } }  } +\sqrt { 1++\cfrac { 1 }{ { 2 }^{ 2 } } +\cfrac { 1 }{ { 3 }^{ 2 } }  } +...+\sqrt { 1++\cfrac { 1 }{ { (2014) }^{ 2 } } +\cfrac { 1 }{ { (2015) }^{ 2 } }  } $$ is