The value of ∑ r = 16 30 r + 2 ( r - 3 ) is equal to:

Given value $=\sum _{r=16}^{30}\left(r+2\right)(r-3)$

$=\sum _{r=1}^{30}\left(r+2\right)(r-3)-\sum _{r=1}^{15}\left(r+2\right)(r-3)$
$=\left(\frac{n\left(n+1\right)\left(2n+1\right)}{6}-\frac{n\left(n+1\right)}{2}-6n\right)-\left(\frac{m\left(m+1\right)\left(2m+1\right)}{6}-\frac{m\left(m+1\right)}{2}-6m\right)$
where $n=30, m=15$
$=8810-1030$
$=7780$