If $\left\{a_{i}\right\}_{i=1}^{n}$ where $n$ is an even integer, is an arithmetic progression with common difference $1$ , and $\sum \limits_{ i =1}^{ n } a _{ i }=192, \sum \limits_{ i =1}^{ n / 2} a _{2 i }=120$, then $n$ is equal to

The sum of the first four terms of an $A.P.$ is $56 .$ The sum of the last four terms is $112.$ If its first term is $11,$ then find the number of terms.

Find the sum to $n$ terms of the $A.P.,$ whose $k^{\text {th }}$ term is $5 k+1$

What is the $20^{\text {th }}$ term of the sequence defined by

$a_{n}=(n-1)(2-n)(3+n) ?$

If $x,y,z$ are in $A.P. $ and ${\tan ^{ – 1}}x,{\tan ^{ – 1}}y$ and ${\tan ^{ – 1}}z$ are also in $A.P.$, then