Question

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

Answer 1

$7 \times 8,10 \times 10,13 \times 12,16 \times 14 \ldots \ldots$
$T _{ n }=(3 n +4)(2 n +6)=2(3 n +4)( n +3)$
$=2\left(3 n ^{2}+13 n +12\right)=6 n ^{2}+26 n +24$
$S _{10}=\displaystyle\sum_{ n =1}^{10} T _{ n }=6 \displaystyle\sum_{ n =1}^{10} n ^{2}+26 \displaystyle\sum_{ n =1}^{10} n +24 \displaystyle\sum_{ n =1}^{10} 1$
$=\frac{6(10 \times 11 \times 21)}{6}+26 \times \frac{10 \times 11}{2}+24 \times 10$
$=10 \times 11(21+13)+240$
$=3980$
Mean $=\frac{ S _{10}}{10}=\frac{3980}{10}=398$