Let ${a_1}, {a_2}, \dots, {a_{49}}$ be in $A.P.$ such that $\sum_{k = 0}^{12} {a_{4k + 1}} = 416$ and ${a_9} + {a_{43}} = 66$. If $\sum_{r = 1}^{17} a_r^2 = 140m$,then $m = \dots$

Jairam purchased a house for Rs. $15000$ and paid Rs. $5000$ at once. He promised to pay the remaining amount in annual installments of Rs. $1000$ with $10\%$ per annum interest on the outstanding balance. How much total money will be paid by Jairam in Rs.?

Consider two sets $A$ and $B$,each containing three numbers in $A.P.$ Let the sum and the product of the elements of $A$ be $36$ and $p$ respectively,and the sum and the product of the elements of $B$ be $36$ and $q$ respectively. Let $d$ and $D$ be the common differences of the $A.P.s$ in $A$ and $B$ respectively such that $D = d + 3$ and $d > 0$. If $\frac{p + q}{p - q} = \frac{19}{5}$,then $p - q$ is equal to:

If $x_1, x_2, \dots, x_n$ and $\frac{1}{h_1}, \frac{1}{h_2}, \dots, \frac{1}{h_n}$ are two $A.P.s$ such that $x_3 = h_2 = 8$ and $x_8 = h_7 = 20$,then $x_5 \cdot h_{10}$ equals

The common difference of the $A.P.$ $b_{1}, b_{2}, \ldots, b_{m}$ is $2$ more than the common difference of $A.P.$ $a_{1}, a_{2}, \ldots, a_{n}$. If $a_{40} = -159$,$a_{100} = -399$ and $b_{100} = a_{70}$,then $b_{1}$ is equal to:

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 $\frac{S_1}{n_1}(n_2 - n_3) + \frac{S_2}{n_2}(n_3 - n_1) + \frac{S_3}{n_3}(n_1 - n_2) = ....$