Question

cho a = 1/2*2+1/3*3+1/4*4+....+1/2017*2017

so sanh a voi 1

 

Answer 1

\(A=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+....+\frac{1}{2017.2017}\)

Ta có :

\(\frac{1}{2.2}< \frac{1}{1.2}\)

\(\frac{1}{3.3}< \frac{1}{2.3}\)

\(\frac{1}{4.4}< \frac{1}{3.4}\)

........

\(\frac{1}{2017.2017}< \frac{1}{2016.2017}\)

=> \(A=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+....+\frac{1}{2017.2017}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{2016.2017}\)

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

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

=> A < 1 

Answer 2

\(a=\frac{1}{2.2}+\frac{1}{3.3}+........+\frac{1}{2017.2017}\)

\(a< \frac{1}{1.2}+\frac{1}{2.3}+......+\frac{1}{2016.2017}\)

\(a< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+......+\frac{1}{2016}-\frac{1}{2017}\)

\(a< 1-\frac{1}{2017}\)

Do \(a< 1-\frac{1}{2017}\)

  \(\Rightarrow a< 1\)