1. Calculate: $1+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+3)+$ $\cdots+\frac{1}{20}(1+2+\cdots+20)$.

$\begin{array}{l}\text { 1. Original expression }=1+\frac{1}{2} \times \frac{2 \times 3}{2}+\frac{1}{3} \times \frac{3 \times 4}{2}+ \\ \cdots+\frac{1}{20} \times \frac{20 \times 21}{2} \\ =\frac{1}{2}(2+3+4+\cdots+21) \\ =\frac{1}{2} \times \frac{23 \times 20}{2}=115 .\end{array}$