Series

The terms are $2, 4, 6, 8, 10$, which add to 30. Equivalently $2 \times \frac{5(6)}{2} = 30$.

Arithmetic with $a=3$, $d=4$: $S_{20} = \frac{20}{2}[6 + 19(4)] = 10(82) = 820$.

$a = 8$ and $r = \tfrac12$, and since $|r| < 1$ the sum is $\dfrac{8}{1-\frac12} = 16$.

$\dfrac{n(n+1)(2n+1)}{6} = \dfrac{10(11)(21)}{6} = \dfrac{2310}{6} = 385$.

$\sum_{r=1}^{10} - \sum_{r=1}^{4} = 55 - 10 = 45$. Check: $5+6+7+8+9+10 = 45$ ✓

$3 \times \dfrac{n(n+1)(2n+1)}{6} - n$. Check at $n = 2$: the terms are 2 and 11, summing to 13, and the formula gives $\dfrac{2(3)(5)}{2} - 2 = 13$ ✓