For $x \geq 0$,the least value of $K$,for which $4^{1+x}+4^{1-x}$,$\frac{K}{2}$,and $16^{x}+16^{-x}$ are three consecutive terms of an $A.P.$ is equal to :

Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}$, where $a > 2$. If $\alpha$ is such that $a, 4, \alpha, b$ are in $A$.$P$.,then the equation $\alpha x^2 - ax + 2(\alpha - 2b) = 0$ has :

The common difference of the $A.P.: a_{1}, a_{2}, ..., a_{m}$ is $13$ more than the common difference of the $A.P.: b_{1}, b_{2}, ..., b_{n}$. If $b_{31} = -277$, $b_{43} = -385$ and $a_{78} = 327$, then $a_{1}$ is equal to

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is $170$. If there are at least $6$ houses in that row and $a$ is the number of the sixth house,then:

What is the sum of all numbers between $100$ and $300$ that are divisible by $7$?

If the sum of the first $n$ terms of the series $\sqrt{3} + \sqrt{75} + \sqrt{243} + \sqrt{507} + \dots$ is $435\sqrt{3}$,then $n$ equals: