Let $x, y > 0$. If $x^{3} y^{2} = 2^{15}$,then the least value of $3x + 2y$ is

$A$ man saves $200$ in each of the first three months of his service. In each of the subsequent months, his saving increases by $40$ compared to the saving of the immediately previous month. His total saving from the start of service will be $11040$ after ............ months.

Statement-$I$: If the ratio of the sum of $n$ terms of two arithmetic progressions is $(7n + 1) : (4n + 17)$,then the ratio of their $n^{th}$ terms is $7 : 4$.
Statement-$II$: If $S_n = an^2 + bn + c$,then $T_n = S_n - S_{n-1}$.

If the average of the first $n$ numbers in the sequence $148, 146, 144, \ldots$ is $125$,then $n$ is equal to

The $20^{th}$ term from the end of the arithmetic progression $3 + 7 + 11 + \dots + 407$ is ...... .

If the sum of $n$ terms of an $A.P.$ is $S_n = nP + \frac{1}{2}n(n-1)Q$,where $P$ and $Q$ are constants,find the common difference.