The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms,the three terms now form an $A.P.$ Then the sum of the original three terms of the given $G.P.$ is

$150$ workers were engaged to finish a piece of work in a certain number of days. $4$ workers dropped out on the second day,$4$ more workers dropped out on the third day,and so on. It takes $8$ more days to finish the work now. The number of days in which the work was completed is:

Find the sum of the first $n$ terms of the $AP$ $\frac{n-1}{n}, \frac{n+1}{n}, \frac{n+2}{n}, \frac{n+3}{n}, \cdots$

Evaluate $\sum_{j=1}^{11} (2 + 3^j)$

If $a^2 + b^2 + 16c^2 = 2(3ab + 6bc + 4ac)$,where $a, b, c$ are non-zero numbers,then $a, b, c$ are in:

The first two terms of a geometric progression add up to $12.$ The sum of the third and the fourth terms is $48.$ If the terms of the geometric progression are alternately positive and negative,then the first term is