Question

Cho A = \(\frac{m-1}{1}+\frac{m-2}{2}+....+\frac{2}{m-2}+\frac{1}{m-1}\)

B = \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{n}\)

Tính A/B

Answer 1

\(A=\frac{m-1}{1}+\frac{m-2}{2}+...+\frac{2}{m-2}+\frac{1}{m-1}\)

\(=\frac{m-1}{1}+\frac{m-2}{2}+...+\frac{m-\left(m-2\right)}{m-2}+\frac{m-\left(m-1\right)}{m-1}\)

\(=m+\frac{m}{2}+\frac{m}{3}+...+\frac{m}{m-1}-1-1-...-1\)

\(=m+\frac{m}{2}+\frac{m}{3}+...+\frac{m}{m-1}-\left(m-1\right)\)

\(=\frac{m}{2}+\frac{m}{3}+...+\frac{m}{m-1}+\frac{m}{m}\)

\(=m\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{m}\right)\)

\(\Rightarrow\frac{A}{B}=m\)