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

The common ratio of a $G.P.$ is $-\frac{4}{5}$ and the sum to infinity is $\frac{80}{9} .$ Find the first term.

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

For $x \in R$,let $[x]$ denote the greatest integer $\le x$. Find the sum of the series $\left[ -\frac{1}{3} \right] + \left[ -\frac{1}{3} - \frac{1}{100} \right] + \left[ -\frac{1}{3} - \frac{2}{100} \right] + \dots + \left[ -\frac{1}{3} - \frac{99}{100} \right]$.

$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{n(n + 1)}$ equals

Three numbers are in $G.P.$ such that their sum is $38$ and their product is $1728$. The greatest number among them is