The $4$th term of an $A.P.$ is equal to $3$ times the first term and the seventh term exceeds twice the third term by $1$. Find the first term and the common difference.

If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that the $4^{\text{th}}$ $A.M.$ is equal to the $2^{\text{nd}}$ $G.M.$,then $m$ is equal to:

Find the sum of the first $7$ terms of a $GP,$ whose first term is $1024$ and the common ratio is $\frac{1}{2}$.

If the $7^{th}$ term of a $H.P.$ is $\frac{1}{10}$ and the $12^{th}$ term is $\frac{1}{25}$,then the $20^{th}$ term is

The sum of $n$ terms of the following series $1 + (1 + x) + (1 + x + x^2) + \dots$ will be

The sum of the first $n$ terms of an $A.P.$ is given by $S_n = 2n + 3n^2$. Another $A.P.$ is formed with the same first term and double the common difference. The sum of the first $n$ terms of this new $A.P.$ is: