Question

\(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}+....+\frac{1}{p_n}< 2\)

Answer 1

\(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}+...+\frac{1}{p_n}\)

Đặt: \(p_1=1.2\)

\(p_2=2.3\)

\(p_3=3.4\)

.....

\(p_n=\left(n-1\right)n\)

\(\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}+...+\frac{1}{p_n}=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right)n}\)

\(\Rightarrow\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{n-1}-\frac{1}{n}\)

\(\Rightarrow\frac{1}{1}-\frac{1}{n}\)

Có \(\frac{1}{1}< 2\Rightarrow\frac{1}{1}-\frac{1}{n}< 2\) (đpcm)