mỗi p/số của A đều bé hơn 1/1.2+1/2.3+1/3.4+......+1/49.50

A<1-1/2+1/2-1/3+1/3-1/4+..........+1/49-1/50(tách ra thành hiệu)

A<1-1/50

mà 1/50>0=>1-1/50<1<2

A<1-1/50<1<2

A<2

chúc học tốt

Đặt \(T=3\cdot5\cdot7\cdot.....\cdot49\)

\(\Rightarrow A\cdot T=\frac{T}{2}+\frac{T}{3}+\frac{T}{4}+....+\frac{T}{50}\)

\(2^4\cdot B\cdot T=\frac{2^4T}{2}+\frac{2^4T}{3}+\frac{2^4T}{4}+....+\frac{2^4T}{50}\left(1\right)\)

Tất cả các số hạng của (1) đều là stn ngoại trừ \(\frac{2^4T}{5}\)

\(\Rightarrow VP\notinℕ\Rightarrow VT\notinℕ\)

Mà \(2^4\inℕ\Rightarrow T\inℕ\)

\(\Rightarrow A\notinℕ\left(đpcm\right)\)

Ta có:\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=4\Rightarrow\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ca}=4\)

\(\Rightarrow2+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ca}=4\Rightarrow\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ac}=2\Rightarrow\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=1\)

\(\Rightarrow\frac{a}{abc}+\frac{b}{abc}+\frac{c}{abc}=1\Rightarrow\frac{a+b+c}{abc}=1\Rightarrow a+b+c=abc\)

\(\Rightarrowđpcm\)

Ta có: \(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{2}\right)^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2}{ab}+\frac{2}{bc}+\frac{2}{ac}\)

\(=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)

\(\Rightarrow2^2=2+2.\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)

\(\Leftrightarrow2=.2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\)

\(\Leftrightarrow\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}=1\)

\(\Leftrightarrow\frac{a}{abc}+\frac{a}{abc}+\frac{b}{abc}=\frac{abc}{abc}\)

\(\Leftrightarrow a+b+c=abc\)

\(\RightarrowĐPCM\)

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=2\)

=> \(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)=4\)

=> \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}=1\)

=> \(\frac{a+b+c}{abc}=1\)

=> a+b+c=abc

A=1+[\(\frac{1}{2^2}+\frac{1}{3^2}+.....+\frac{1}{50^2}\)

ta có \(\frac{1}{2^2}<\frac{1}{1.2};\frac{1}{3^2}<\frac{1}{2.3};......;\frac{1}{50^2}<\frac{1}{49.50}\)

=>A<1+\(\left[\frac{1}{1.2}+.........+\frac{1}{49.50}\right]\)

=>A<1+\(\left[\frac{1}{1}-\frac{1}{50}\right]\)

=>A<1+\(\frac{49}{50}\)

=>A<\(\frac{99}{50}\) <2

=>A<2

K MÌNH NHA BÀI NÀY MÌNH GHI MỎI TAY LẮM

A=\(\frac{1}{1^2}+\frac{1}{2^2}+....+\frac{1}{50^2}\)

A<\(1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49\cdot50}\)

A<1+\(\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\right)\)

A<1+\(\left(1-\frac{1}{50}\right)\)

A<1+\(\frac{49}{50}\)

=>A<2

Ta có : \(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2015^2}=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{2015.2015}\) 

\(< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2014.2015}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2015}\)

\(=1-\frac{1}{2015}=\frac{2014}{2015}< 1\)

=> A < 1 (đpcm)

cho 1/a+1/b+1/c=2  va :a+b+c=abc

.chung minh rang: 

.

Ta có A = 1/2 + 1/2^2 + 1/2^3 + ... + 1/2^100

Suy ra 2A - A = ( 1 + 1/2 + 1/2^2 +...+ 1/2^99) - ( 1/2 + 1/2^2 +...+ 1/2^100 )

Suy ra A = 1 - 1/2^100 < 1

Vậy A < 1 ( ĐPCM)