If $S_1, S_2$ and $S_3$ are the sums of the first $n_1, n_2$ and $n_3$ terms of an arithmetic progression respectively,then $\frac{S_1}{n_1}(n_2 - n_3) + \frac{S_2}{n_2}(n_3 - n_1) + \frac{S_3}{n_3}(n_1 - n_2) = ....$

When the $9^{th}$ term of an $A.P.$ is divided by its $2^{nd}$ term,the quotient is $5$. When the $13^{th}$ term is divided by the $6^{th}$ term,the quotient is $2$ and the remainder is $5$. Find the first term of the $A.P.$

If $a, b, c, d, e$ are in an arithmetic progression,then what is the value of $a - 4b + 6c - 4d + e$?

If ${a_1}, {a_2}, ..., {a_n}$ are positive real numbers whose product is a fixed number $c$,then the minimum value of ${a_1} + {a_2} + ... + {a_{n-1}} + 2{a_n}$ 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 $40$ more than the saving of the immediately previous month. His total saving from the start of service will be $11040$ after ............ months.

If $a_1, a_2, \dots, a_n$ are positive real numbers such that their product is a constant $c$,then what is the minimum value of $a_1 + a_2 + \dots + a_{n-1} + 2a_n$?