If the average of the first $n$ numbers in the sequence $148, 146, 144, \ldots$ is $125$,then $n$ is equal to

If $a_1, a_2, a_3, . . . , a_n, . . .$ are in $A.P.$ such that $a_4 - a_7 + a_{10} = m$,then the sum of the first $13$ terms of this $A.P.$ is .............. $m$.

If the first term and the last term of an arithmetic progression are $a$ and $\ell$ respectively,and the sum of all its terms is $S$,then what is its common difference $d$?

$A$ software company sets up $m$ number of computer systems to finish an assignment in $17$ days. If $4$ computer systems crashed on the start of the second day,$4$ more computer systems crashed on the start of the third day and so on,then it took $8$ more days to finish the assignment. The value of $m$ is equal to :

If three positive numbers $a, b,$ and $c$ are in $A.P.$ such that $abc = 8$,then the minimum possible value of $b$ is

The sum of the first $p, q,$ and $r$ terms of an $A.P.$ are $a, b,$ and $c,$ respectively. Prove that $\frac{a}{p}(q-r)+\frac{b}{q}(r-p)+\frac{c}{r}(p-q)=0$.