Question

CMR:1/2²+1/3²+1/4²+...+1/100²<1

Answer 1

Tổng quát: \(\frac{1}{n^2}<\frac{1}{\left(n-1\right).n}\)

Ta có: \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{100^2}<\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{99.100}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}=1-\frac{1}{100}=\frac{99}{100}<1\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}<1\)

Answer 2

Đặt biểu thức ở vế trái là A ta có

\(A<\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}=\)

\(=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{100-99}{99.100}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}=\)

\(=1-\frac{1}{100}<1\Rightarrow A<1\) (dpcm)