If $a, b,$ and $c$ are positive real numbers,then the minimum value of $(a + b + c)(1/a + 1/b + 1/c)$ is .......

Let $T_r$ be the $r^{\text{th}}$ term of an $A.P.$ If for some $m$,$T_m = \frac{1}{25}$,$T_{25} = \frac{1}{20}$ and $20 \sum_{r=1}^{25} T_r = 13$,then $5m \sum_{r=m}^{2m} T_r$ is equal to:

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 $m = \dots$

If the ratio of the sum of $n$ terms of two arithmetic progressions is $(7n + 1) : (4n + 27)$,then what is the ratio of their $11^{th}$ terms?

Consider an $A.P.$ of positive integers,whose sum of the first three terms is $54$ and the sum of the first twenty terms lies between $1600$ and $1800$. Then its $11^{\text{th}}$ term 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.