Find the sum of $1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots$ to infinite terms.

The difference between an integer and its cube is divisible by:

Let the sum of the first $n$ terms of a non-constant $A.P., a_1, a_2, a_3, \dots$ be $S_n = 50n + \frac{n(n - 7)}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.,$ then the ordered pair $(d, a_{50})$ is equal to

The sum of the series $1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \dots$ up to $10$ terms is:

The average of $n$ numbers is $a$. The first number is increased by $2$,the second one is increased by $4$,the third one is increased by $8$,and so on. The average of the new numbers is:

The first term of a $G.P.$ is $7$,the last term is $448$,and the sum of all terms is $889$. Then the common ratio is: