In an $AP$ given $a=3, n=8, S_n=192,$ find $d$.

For what value of $n$,are the $n^{th}$ terms of two $APs: 63, 65, 67, \ldots$ and $3, 10, 17, \ldots$ equal?

$A$ 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 $1^{st}$ year.

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

In an $AP$,given $l = 28$,$S = 144$,and there are a total of $9$ terms. Find $a$.

How many terms of the $AP: 24, 21, 18, \ldots$ must be taken so that their sum is $78$?