$A$ manufacturer of $TV$ sets produced $600$ sets in the third year and $700$ sets in the seventh year. Assuming that the production increases uniformly by a fixed number every year,find the production in the $10^{th}$ year.

If the sum of the first $14$ terms of an $AP$ is $1050$ and its first term is $10,$ find the $20^{th}$ term.

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

Check whether $301$ is a term of the list of numbers $5, 11, 17, 23, \ldots$

Write the first four terms of the $AP,$ when the first term $a$ and the common difference $d$ are given as follows: $a = 10, d = 10.$

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 $x$.