If $a,b,c$ are in $A.P.$, then $\frac{1}{{\sqrt a + \sqrt b }},\,\frac{1}{{\sqrt a + \sqrt c }},$ $\frac{1}{{\sqrt b + \sqrt c }}$ are in

If ${S_k}$ denotes the sum of first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively, then ${S_{kn}}/{S_n}$ be independent of $n$ if

Let $a_{1}, a_{2} \ldots, a_{n}$ be a given $A.P.$ whose common difference is an integer and $S _{ n }= a _{1}+ a _{2}+\ldots+ a _{ n }$ If $a_{1}=1, a_{n}=300$ and $15 \leq n \leq 50,$ then the ordered pair $\left( S _{ n -4}, a _{ n -4}\right)$ is equal to

If the ${9^{th}}$ term of an $A.P.$ is $35$ and ${19^{th}}$ is $75$, then its ${20^{th}}$ terms will be

The sides of a right angled triangle are in arithmetic progression. If the triangle has area $24$ , then what is the length of its smallest side ?