Question

[Hồ Văn Minh Nhật]

Chứng minh:  1/1.6+1/6.11+...+1/(5n+1)(5n+6)=n+1/5n+6

Answer 1

[Akai Haruma]

Lời giải:

$A=\frac{1}{1.6}+\frac{1}{6.11}+....+\frac{1}{(5n+1)(5n+6)}$

$5A=\frac{6-1}{1.6}+\frac{11-6}{6.11}+....+\frac{(5n+6)-(5n+1)}{(5n+1)(5n+6)}$

$5A=1-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+....+\frac{1}{5n+1}-\frac{1}{5n+6}$

$=1-\frac{1}{5n+6}=\frac{5n+5}{5n+6}$

$\Rightarrow A=\frac{n+1}{5n+6}$