Tính E = 1/1.7 + 1/7.13 + 1/13.19 +...+ 1/31.37

Đề bài: Tính $E=\frac{1}{1.7}+\frac{1}{7.13}+\frac{1}{13.19}+\mathrm{...}+\frac{1}{31.37}$ .

Trả lời

Hướng dẫn giải:

$$\begin{gathered} E=\frac{1}{1.7}+\frac{1}{7.13}+\frac{1}{13.19}+\mathrm{...}+\frac{1}{31.37} \\ E=\frac{1}{6}\left(\frac{6}{1.7}+\frac{6}{7.13}+\frac{6}{13.19}+\mathrm{...}+\frac{6}{31.37}\right) \\ E=\frac{1}{6}\left(1-\frac{1}{7}+\frac{1}{7}-\frac{1}{13}+\frac{1}{13}-\frac{1}{19}+\mathrm{...}+\frac{1}{31}-\frac{1}{37}\right) \\ E=\frac{1}{6}\left(1-\frac{1}{37}\right)=\frac{1}{6}.\frac{36}{37}=\frac{6}{37} \end{gathered}$$

Vậy $E=\frac{6}{37}$ .