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 $x$.

Find the number of terms in the following $AP$: $18, 15 \frac{1}{2}, 13, \ldots, -47$.

Do $4, 10, 16, 22, \ldots$ form an $AP$? If they form an $AP,$ write the next two terms.

Find the sum of:
$(i)$ the first $1000$ positive integers
$(ii)$ the first $n$ positive integers

Find the sum of the first $51$ terms of an $AP$ whose second and third terms are $14$ and $18$ respectively.