The first term of an $AP$ is $5,$ the last term is $45$ and the sum is $400.$ Find the number of terms and the common difference.

$y$ years ago,the sum of the ages of five friends was $x$ years,then the sum of the present ages of them is $\ldots \ldots \ldots \ldots$ years.

Jaspal Singh repays his total loan of $Rs. 118000$ by paying every month starting with the first instalment of $Rs. 1000$. If he increases the instalment by $Rs. 100$ every month,what amount will be paid by him in the $30^{\text{th}}$ instalment? What amount of loan does he still have to pay after the $30^{\text{th}}$ instalment?

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 $m$)

The eighth term of an $AP$ is half its second term and the eleventh term exceeds one third of its fourth term by $1$. Find the $15^{\text{th}}$ term.

With respect to the usual notations of an $A.P.$,if $T_{n} = 4$,$d = 2$,and $S_{n} = -14$,find $n$ and $a$.