Cho A = 1/2.3 + 1/3.4 + 1/4.5 + ... + 1/99.100. So sánh A với 1/2

* Hướng dẫn giải

Ta có $\text{A}=\left(\frac{\text{1}}{\text{2}}-\frac{\text{1}}{\text{3}}\right)+\left(\frac{\text{1}}{\text{3}}-\frac{\text{1}}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+\mathrm{...}+\left(\frac{\text{1}}{\text{99}}-\frac{\text{1}}{\text{100}}\right)$

$\begin{array}{l}A=\frac{1}{2}+\left(-\frac{1}{3}+\frac{1}{3}\right)+\left(-\frac{1}{4}+\frac{1}{4}\right)+\left(-\frac{1}{5}+\frac{1}{5}\right)+\mathrm{...}+\left(-\frac{1}{99}+\frac{1}{99}\right)-\frac{1}{100}\\ A=\frac{1}{2}-\frac{1}{100}=\frac{49}{100}<\frac{50}{100}\end{array}$

Vậy $A<\frac{1}{2}.$