Find the sum of the first $20$ terms of the $A.P.$ $5, 13, 21, \ldots$ and find the number of terms of this $A.P.$ whose sum is $6440$.

In an $AP$,given $a=2, d=8, S_{n}=90$,find $n$ and $a_{n}$.

The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. They have $27$ flags to be fixed at intervals of every $2 \,m$. The flags are stored at the position of the middle-most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much total distance did she cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag? (in $m$)

In an $AP$ given $a_{12}=37$ and $d=3$,find the first term $a$ and the sum of the first $12$ terms $S_{12}$.

The sum of three numbers in $A.P.$ is $21$. The product of the first and the third number exceeds the second number by $6$. Find those numbers.

With respect to the usual notations of an $A.P.$,if $a=5, d=3$ and $T_n=50$,find $n$ and $S_n$.