Question

[Xin chào]

chứng minh A = 1/1.2+1/2.3+1/3.4+....+1/n(n+1)=n/n+1 (n thuộc n*)

Answer 1

[Akai Haruma]

Lời giải:

$A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{n(n+1)}$

$=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+....+\frac{(n+1)-n}{n(n+1)}$

$=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n}-\frac{1}{n+1}$

$=1-\frac{1}{n+1}=\frac{n}{n+1}$
Ta có đpcm.