A) 95
B) 110
C) 115
D) 155
Correct Answer: C
Solution :
[c] \[\sum\limits_{i=1}^{20}{i}\,\left( \frac{1}{i}+\frac{1}{i+1}+\frac{1}{i+2}+...+\frac{1}{20} \right)\] \[=1.\left( \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{20} \right)+2.\left( \frac{1}{2}+\frac{1}{3}+...+\frac{1}{20} \right)\] \[+3.\left( \frac{1}{3}+...+\frac{1}{20} \right)+...+20.\left( \frac{1}{20} \right)\] \[=\frac{1}{1}(1)+\frac{1}{2}(1+2)+\frac{1}{3}(1+2+2)+...\] \[+\frac{1}{20}(1+2+3+...+20)\] \[=\frac{1+1}{2}+\frac{2+1}{2}+\frac{3+1}{2}+...+\frac{20+1}{2}=\frac{20\times 21}{4}+10\]\[=115\]
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