In an $AP$,given $a=8, a_{n}=62, S_{n}=210$,find $n$ and $d$.

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?

The first term of an $AP$ is $5,$ the last term is $45$ and the sum is $400.$ Find the number of terms and the common difference.

Find the sum of the first $24$ terms of the list of numbers whose $n^{th}$ term is given by $a_{n} = 3 + 2n$.

How many terms of the $AP : 9, 17, 25, \ldots$ must be taken to give a sum of $636$?

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 $m$)