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

The taxi fare after each $km$ when the fare is ₹ $15$ for the first $km$ and ₹ $8$ for each additional $km$. Does the list of numbers involved form an arithmetic progression,and why?

$A$ contract on a construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹ $200$ for the first day,₹ $250$ for the second day,₹ $300$ for the third day,etc. The penalty for each succeeding day is ₹ $50$ more than for the preceding day. How much money does the contractor have to pay as a penalty if he has delayed the work by $30$ days?

For the $AP: \frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}, \ldots,$ write the first term $a$ and the common difference $d$.

$A$ 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.

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