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 houses of a row are numbered consecutively from $1$ to $49$. Show that there is a value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it. Find this value of $x$.

For the following $APs$,write the first term and the common difference: $\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \ldots$

Find the $10^{th}$ term of the $AP : 2, 7, 12, \ldots$

If the sum of the first $n$ terms of an $AP$ is $4n - n^2$,what is the first term (that is $S_1$)? What is the sum of first two terms? What is the second term? Similarly,find the $3^{rd}$,the $10^{th}$ and the $n^{th}$ terms.

In a potato race,a bucket is placed at the starting point,which is $5\, m$ from the first potato,and the other potatoes are placed $3\, m$ apart in a straight line. There are ten potatoes in the line. $A$ competitor starts from the bucket,picks up the nearest potato,runs back with it,drops it in the bucket,runs back to pick up the next potato,runs to the bucket to drop it in,and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run? (in $m$)