Question

Cho:

A=1/1.2+1/2.3+1/3.4+....+1/2016.2017.Chứng tỏ A<1

Answer 1

A = \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2016.2017}\)

A = \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2016}-\frac{1}{2017}\)

A = \(1-\left(\frac{1}{2}-\frac{1}{2}\right)-\left(\frac{1}{3}-\frac{1}{3}\right)-...-\left(\frac{1}{2016}-\frac{1}{2016}\right)-\frac{1}{2017}\)

A = \(1-0-0-0...-0-\frac{1}{2017}\)

A = \(1-\frac{1}{2017}< 1\)

Answer 2

\(A=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2016.2017}\)

\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2016}-\frac{1}{2017}\)

\(A=1-\frac{1}{2017}=\frac{2016}{2017}< \frac{2017}{2017}=1\)

=> A<1(đpcm)

Answer 3

1/1×2+1/2×3+1/3×4+...+1/2016×2017 =1/1-1/2+1/2-1/3+1/3-1/4+...+1/2016-1/2017 =1/1-1/2017<1 =>A<1 Vậy A<1