Find the sum of all integers between $50$ and $500$ which are divisible by $7.$

$A$ person is to count $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \ldots = a_{10} = 150$ and $a_{10}, a_{11}, \ldots$ are in an $A.P.$ with common difference $-2$,then the time taken by him to count all notes is ............... minutes.

The first term of an $A.P.$ is $2$ and the common difference is $4$. The sum of its $40$ terms will be:

Let $a_{n}$ be the $n^{\text{th}}$ term of a $G$.$P$. of positive terms. If $\sum_{n=1}^{100} a_{2n+1} = 200$ and $\sum_{n=1}^{100} a_{2n} = 100$,then $\sum_{n=1}^{200} a_{n}$ is equal to:

The harmonic mean of $\frac{a}{1 - ab}$ and $\frac{a}{1 + ab}$ is

The ratio of the $7^{th}$ to the $3^{rd}$ term of an $A.P.$ is $12:5$. Find the ratio of the $13^{th}$ to the $4^{th}$ term.