Question

The mean of $10$ numbers $7\times 8,10\times 10,13\times 12,16\times 14,....$ is

Answer 1

$S=7\times 8+10\times 10+13\times 12+\cdots 10\text{term}$
The mean of the numbers is
$\frac{S}{10}=\frac{1}{10}\sum _{r=1}^{10}(3r+4)(2r+6)\Rightarrow \frac{S}{10}=\frac{2}{10}\sum _{r=1}^{10}(3r^2+13r+12)\Rightarrow \frac{S}{10}=\frac{2}{10}(3\times \frac{10\times 11\times 21}{6}+13\times \frac{10\times 11}{2}+12\times 10)\Rightarrow \frac{S}{10}=2(3\times \frac{11\times 7}{2}+13\times \frac{11}{2}+12)\Rightarrow \frac{S}{10}=2(187+12)\therefore \frac{S}{10}=398$