Question

[Phan Ngọc Anh]

CMR:

1/2-1/4+1/8-1/116+1/32-1/64<1/3

GIÚP MIK NHA

XIN CẢM ƠN

Answer 1

[Lê Nguyệt Hằng]

Đặt A= \(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)

         =\(\frac{1}{2^1}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^6}\)

=> 2A= \(1-\frac{1}{2^1}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}\)

Ta có:

2A+A=\(\left(1-\frac{1}{2^1}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}\right)+\left(\frac{1}{2^1}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^6}\right)\)

=> 3A=\(1-\frac{1}{2^1}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}+\frac{1}{2^1}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^6}\)

=\(1-\left(\frac{1}{2^1}-\frac{1}{2^1}\right)+\left(\frac{1}{2^2}-\frac{1}{2^2}\right)-\left(\frac{1}{2^3}-\frac{1}{2^3}\right)+\left(\frac{1}{2^4}-\frac{1}{2^4}\right)-\left(\frac{1}{2^5}-\frac{1}{2^5}\right)-\frac{1}{2^6}\)

= \(1-\frac{1}{2^6}\)

=> A= 3A:3= \(\left(1-\frac{1}{2^6}\right):3\)=\(\frac{1}{3}-\frac{1}{2^6}:3\)<\(\frac{1}{3}\)