Determine $k$ so that $k^{2}+4k+8, 2k^{2}+3k+6, 3k^{2}+4k+4$ are three consecutive terms of an $AP$.

Find the sum of the first $25$ terms of the $A.P.$ $30, 26, 22, \ldots$

Yasmeen saves $Rs.\, 32$ during the first month,$Rs.\, 36$ in the second month,and $Rs.\, 40$ in the third month. If she continues to save in this manner,in how many months will she save $Rs.\, 2000$?

For the finite $A.P.$ $12, 21, 30, \ldots, 363,$ find the $12^{th}$ term from the end.

The $n^{th}$ term of an $A.P.$ is given by $T_{n} = 3n - 1$. Then,the common difference of the $A.P.$ is ...........

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 $m$)