Sequences and S...

Value of $$9 + 99 + 999 + .... $$ upto $$n$$ terms is

The sum of the series $$4 + 8 + 16 + 32 + .......$$. till $$10$$ terms is

$$\displaystyle \sum _{ p=1 }^{ 32 }{ \left( 3p+2 \right) { \left[ \sum _{ q=1 }^{ 10 }{ \left( \sin { \frac { 2q\pi  }{ 11 }  } -i\cos { \frac { 2q\pi  }{ 11 }  }  \right)  }  \right]  }^{ p } } =$$

Find the sum the infinite G.P.:
$$1\, +\, \displaystyle {\frac{1}{3}\, +\, \frac{1}{9}\, +\, \frac{1}{27}\, +\, .......}$$

Evaluate $$7 + 77 + 777 + .............$$ upto $$n$$ terms.

For $$\frac {2^2+4^2+6^2+...+(2n)^2}{1^2+3^2+5^2+...+(2n-1)^2}$$ to exceed 1.01, the maximum value of n is

In  the following question, the numbers/letters are arranged based on some pattern or principle.Choose the correct answer for the term marked by the symbol (?) 

The sum of 'n' terms of series $$1^2+(1^2+2^2)+(1^2+ 2^2+ 3^2)+(1^2+2^2+ 3^2+4^2)+ .......$$ will be

$$11, 26, 56, 101, 161, .....$$

The value of $$\displaystyle\frac{1}{1\cdot2\cdot3}+\displaystyle\frac{1}{2\cdot3\cdot4}+\displaystyle\frac{1}{3\cdot4\cdot5}+\displaystyle\frac{1}{4\cdot5\cdot6}$$ is equal to