If the $10^{\text{th}}$ term of an $A$.$P$. is $\frac{1}{20}$ and its $20^{\text{th}}$ term is $\frac{1}{10}$,then the sum of its first $200$ terms is

$150$ workers were engaged to finish a job 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 took $8$ more days to finish the work. Find the number of days in which the work was completed.

Let $T_r$ be the $r^{th}$ term of an $A.P.$ for $r = 1, 2, 3, \dots$. If for some positive integers $m, n$ we have $T_m = \frac{1}{n}$ and $T_n = \frac{1}{m}$,then $T_{mn}$ equals

The sequence $\log a, \log \frac{a^2}{b}, \log \frac{a^3}{b^2}, \ldots$ is

Let $x_n, y_n, z_n, w_n$ denote the $n^{th}$ terms of four different arithmetic progressions with positive terms. If $x_4 + y_4 + z_4 + w_4 = 8$ and $x_{10} + y_{10} + z_{10} + w_{10} = 20$,then the maximum value of $x_{20} \cdot y_{20} \cdot z_{20} \cdot w_{20}$ is:

Let $\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}$ be an $A$.$P$. of four terms such that each term of the $A$.$P$. and its common difference $l$ are integers. If $\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=48$ and $\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}+l^{4}=361$, then the largest term of the $A$.$P$. is equal to