Sequences and S...

The value of $$\left(\displaystyle 1-\frac{1}{2}\right)\left(\displaystyle 1-\frac{1}{3}\right)\left(\displaystyle 1-\frac{1}{4}\right).........\left(\displaystyle 1-\frac{1}{10}\right)=$$ ___________.

Sum $$1+3x+5{ x }^{ 2 }+7{ x }^{ 3 }+9{ x }^{ 4 }+.....$$ to infinity is $$\left( x<1 \right) $$

$$1 \, + \, 2 \, \cdot \, 2 \, + \, 3 \, \cdot \, 2^2 \, + \, 4 \, \cdot \, 2^3 \, + \, ..... \, + \, 100 \, \cdot \, 2^{99}$$

Select the missing number from the given matrix:

The two sequence of number $${1,4,16,64,.......}$$ and $${3,12,48,192,........}$$ are mixed as follows:$${1,3,4,12,16,48,64,192,.......}$$. one of the numbers in the mixed in the mixed series is $$1048576$$, then number immediately preceding is-

If $$\dfrac{a-x}{px}=\dfrac{a-y}{qy}=\dfrac{a-z}{rz}$$ and $$p, q, r,$$ be in A.P. then $$x, y, z$$ are in?

What is the next term of the AP $$\sqrt{2}, \sqrt{8}, \sqrt{18},$$ ____?

$$A$$ is twice as fast as $$B$$ is thrice as fast as $$C$$. The journey covered by $$C$$ in $$42$$ minutes, what will be covered by $$A$$ is 

The sum of the infinite series $$1+\left ( 1+\dfrac{1}{5} \right )\left ( \dfrac{1}{2} \right )+\left ( 1+\dfrac{1}{5}+\dfrac{1}{5^2} \right )\left ( \dfrac{1}{2^2} \right )+.......$$

The sum of the series $$\sum_{r=0}6{88} \sec r^o\cdot \sec(r+1)^o$$ is equal to