Tính bằng cách hợp lí: C = 2/1.4 + 2/4.7 + 2/7.9 + ... + 2/196.199

* Hướng dẫn giải

$C=\frac{\text{2}}{\text{1.4}}+\frac{\text{2}}{\text{4.7}}+\frac{\text{2}}{\text{7.9}}+\mathrm{...}+\frac{\text{2}}{\text{196.199}}$

$C=\frac{\text{2}}{\text{1.4}}+\frac{\text{2}}{\text{4.7}}+\frac{\text{2}}{\text{7.9}}+\mathrm{...}+\frac{\text{2}}{\text{196.199}}$

$C=\frac{2}{3}.\left[\left(1-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{7}\right)+\left(\frac{1}{7}-\frac{1}{9}\right)+\mathrm{...}+\left(\frac{1}{196}-\frac{1}{199}\right)\right]$

$C=\frac{2}{3}.\left[1+\left(-\frac{1}{4}+\frac{1}{4}\right)+\left(-\frac{1}{7}+\frac{1}{7}\right)+\left(-\frac{1}{9}+\frac{1}{9}\right)+\mathrm{...}+\left(-\frac{1}{196}+\frac{1}{196}\right)-\frac{1}{199}\right]$

$C=\frac{2}{3}.\left(1-\frac{1}{199}\right)=\frac{2}{3}.\frac{198}{199}=\frac{132}{199}.$