The number of terms common to the two A.P.'s $3,7,11, \ldots ., 407$ and $2,9,16, \ldots . .709$ is

$150$ workers were engaged to finish a piece of work in a certain number of days. $4$ workers dropped the second day, $4$ more workers dropped the third day and so on. It takes eight more days to finish the work now. The number of days in which the work was completed is

If in the equation $a{x^2} + bx + c = 0,$ the sum of roots is equal to sum of square of their reciprocals, then $\frac{c}{a},\frac{a}{b},\frac{b}{c}$ are in

Write the first five terms of the sequences whose $n^{t h}$ term is $a_{n}=(-1)^{n-1} 5^{n+1}$

There are $15$ terms in an arithmetic progression. Its first term is $5$ and their sum is $390$. The middle term is

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