The $3^{rd}$ term of a $G.P.$ is the square of the first term. If the second term is $8,$ determine the $6^{th}$ term.

Find the sum of an $AP$ of $40$ terms whose first and last terms are $80$ and $275$.

For all positive integral values of $n$,the value of $3 \cdot 1 \cdot 2 + 3 \cdot 2 \cdot 3 + 3 \cdot 3 \cdot 4 + \dots + 3 \cdot n \cdot (n + 1)$ is

$150$ workers were engaged to finish a piece of work in a certain number of days. $4$ workers dropped out on the second day,$4$ more workers dropped out on the third day,and so on. It takes $8$ more days to finish the work now. The number of days in which the work was completed is:

If the ${5^{th}}$ term of a $H.P.$ is $\frac{1}{45}$ and the ${11^{th}}$ term is $\frac{1}{69}$,then its ${16^{th}}$ term will be:

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an $A.P.$ are $a, b,$ and $c$ respectively,find the value of $a(q-r) + b(r-p) + c(p-q)$.