Sequences and S...

If $$\displaystyle \sum_{s=1}^{n}\, \left \{ \displaystyle \sum_{r=1}^{s}r \right \}\, =\, an^3\, +\, bn^2\, +\, cn$$, then find the value of a + b + c.

The sum of n terms of the series
$$1 + (1 + a) + (1 + a + a^{2}) + (1 + a + a^{2} + a^{3}) + .....$$, is

The sum of series $$\displaystyle \frac {1^2}{1}+\frac {1^2+2^2}{1+2}+\frac {1^2+2^2+3^2}{1+2+3}+....$$ upto n terms is

$$\dfrac {1^3+2^3+3^3+4^3+........12^3}{1^2+2^2+3^2+4^2+......12^2}=$$

1 + 3 + 6 + 10 + .... upto n terms is equal to

The value of $$\displaystyle \sum_{n=1}^{10}\left (\sin \dfrac {2n\pi}{11}-\cos\dfrac {2n\pi}{11}\right )$$ is equal to

The sum of the first n terms of the series $$\displaystyle 1^{2}+2.2^{2}+3^{3}+2.4^{2}+5^{2}+2.6^{2}+....$$ is $$\displaystyle \frac{n\left ( n+1 \right )^{2}}{2}$$ when n is even. When n is odd the sum is

The infinite sum $$1\, +\, \displaystyle {\frac{4}{7}\, +\, \frac{9}{7^2}\, +\, \frac{16}{7^3}\, +\, \frac{25}{7^4}\, +\, ......}$$ equals

If $$\displaystyle \sum_{r=1}^{n}t_{r}=\frac{1}{12}n\left ( n+1 \right )\left ( n+2 \right )$$, then the value of $$\displaystyle \sum_{r=1}^{n}\frac{1}{t_{r}}$$ is

If $$\displaystyle x_{i}> 0,i=1,2,....,50\: \: and\: \: x_{1}+x_{2}+...+x_{50}=50 $$ then the minimum value of $$\displaystyle \frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{50}}$$ equals to