The value of the sum ∑i=120 i1i+1i+1+1i+2+…+120 is

# The value of the sum $\sum _{i=1}^{20} i\left[\frac{1}{i}+\frac{1}{i+1}+\frac{1}{i+2}+\dots +\frac{1}{20}\right]$ is

1. A

100

2. B

105

3. C

110

4. D

115

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### Solution:

$\begin{array}{l}\sum _{i=1}^{20} i\left(\frac{1}{i}+\frac{1}{i+1}+\frac{1}{i+2}+\dots +\frac{1}{20}\right)\\ =\left(\frac{1}{1}+\frac{1}{2}+\cdots +\frac{1}{20}\right)+2\left(\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{20}\right)+3\left(\frac{1}{3}+\cdots +\frac{1}{20}\right)+20\left(\frac{1}{20}\right)\\ =\left(1\right)\left(1\right)+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+\cdots +\frac{1}{20}\left(1+2+\cdots +20\right)\end{array}$$\begin{array}{l}=\sum _{k=1}^{20} \frac{1}{k}\left(1+2+\cdots +k\right)=\frac{1}{2}\sum _{k=1}^{20} \left(k+1\right)\\ =\frac{1}{2}\cdot \frac{1}{2}\left(20\right)\left(2+21\right)=115\end{array}$