Sequences and S...

The value of the infinite series $$\cfrac{1^2+2^2}{\underline{|3}} + \cfrac{1^2+2^2+3^2}{\underline{|4}}+\cfrac{1^2+2^2+3^3+4^2}{\underline{|5}}........$$ is

Let $$S$$ denote the sum of the infinite series $$1+\cfrac { 8 }{ 2! } +\cfrac { 21 }{ 3! } +\cfrac { 40 }{ 4! } +\cfrac { 65 }{ 5! } +.......$$. Then

$$x=1+\frac{1}{2\times \underline{|1}}+\frac{1}{4\times \underline{|2}}+\frac{1}{8\times \underline{|3}}$$

In the series $$1+3+6+10+......,$$ find the $$n^{th}$$ term:

Consider an incomplete pyramid of balls on a square base having $$18$$ layers; and having $$13$$ balls on each side of the top layer. Then the total number $$N$$ of balls in that pyramid satisfies

With the help of match-sticks, Zalak prepared a pattern as shown below. When $$97$$ matchsticks are used, the serial number of the figure will be ...........

Find the $$(2n)^{th}$$ term of the series whose $$n^{th}$$ term is $$\dfrac{n^2+1}{n^3}$$:

The value of $$\dfrac {1}{i} + \dfrac {1}{i^{2}} + \dfrac {1}{i^{3}} + .... + \dfrac {1}{i^{102}}$$ is

The sum of the series $$\displaystyle \sum_{n = 8}^{17} \dfrac{1}{(n + 2)(n + 3)} $$ is equal to

Let $$a_n=\dfrac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}$$ then $$\displaystyle \sum_{n=1}^{144}a_n$$ equals