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

Let $S_n$ be the sum of all integers $k$ such that $2^n < k < 2^{n+1}$,for $n \geq 1$. Then,$9$ divides $S_n$ if and only if

After inserting $n$ arithmetic means $(A.M.s)$ between $2$ and $38$,the sum of the resulting progression is $200$. The value of $n$ is

If the sum of three numbers in an arithmetic sequence is $15$ and the sum of their squares is $83$,then the numbers are

Let $x_n, y_n, z_n, w_n$ denote the $n^{th}$ terms of four different arithmetic progressions with positive terms. If $x_4 + y_4 + z_4 + w_4 = 8$ and $x_{10} + y_{10} + z_{10} + w_{10} = 20$,then the maximum value of $x_{20} \cdot y_{20} \cdot z_{20} \cdot w_{20}$ is:

Find the sum of integers from $1$ to $100$ that are divisible by $2$ or $5.$