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Bài 1 :
\(A=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{50-49}{49.50}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)
\(=1-\frac{1}{50}< 1\left(1\right)\)
\(B=\frac{1}{10}+\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{99}+\frac{1}{100}\right)\)\(>\frac{1}{10}+\frac{1}{100}.90=1\left(2\right)\)
Từ (1) và ( 2) ta có \(A< 1\) \(B>1\)NÊN \(A< B\)
Bài 2:
\(S=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\)
\(=\frac{\left(a+b+c\right)-\left(b+c\right)}{b+c}+\)\(\frac{\left(a+b+c\right)-\left(c+a\right)}{c+a}\)\(+\frac{\left(a+b+c\right)-\left(a+b\right)}{a+b}\)
\(=\frac{7-\left(b+c\right)}{b+c}+\frac{7-\left(c+a\right)}{c+a}+\frac{7-\left(a+b\right)}{a+b}\)
\(=7.\left(\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}\right)-3\)
\(=7.\frac{7}{10}-3\)\(=\frac{49}{10}-3=\frac{19}{10}\)
\(S=\frac{19}{10}>\frac{19}{11}=1\frac{8}{11}\)
Chúc bạn học tốt ( -_- )
Bài 1:
ta có: \(A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)
\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)
\(A=1-\frac{1}{50}< 1\)
\(\Rightarrow A< 1\)(1)
ta có: \(\frac{1}{11}>\frac{1}{100};\frac{1}{12}>\frac{1}{100};...;\frac{1}{99}>\frac{1}{100}\)
\(\Rightarrow\frac{1}{11}+\frac{1}{12}+...+\frac{1}{99}+\frac{1}{100}>\frac{1}{100}+\frac{1}{100}+...+\frac{1}{100}+\frac{1}{100}\) ( có 90 số 1/100)
\(=\frac{90}{100}=\frac{9}{10}\)
\(\Rightarrow B=\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+...+\frac{1}{99}+\frac{1}{100}>\frac{1}{10}+\frac{9}{10}=1\)
\(\Rightarrow B>1\)(2)
Từ (1);(2) => A<B
Xét
\(\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)=7\cdot\frac{7}{10}=\frac{49}{10}\)
\(\Leftrightarrow\frac{a+b}{a+b}+\frac{c}{a+b}+\frac{a+c}{a+c}+\frac{b}{a+c}+\frac{b+c}{b+c}+\frac{a}{b+c}=\frac{49}{10}\)
\(3+\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=\frac{49}{10}\Leftrightarrow S=\frac{19}{10}\)
Ta có: \(1\frac{8}{11}=\frac{19}{11}\)
vì 19=19 ,\(\frac{1}{11}< \frac{1}{10}\)nên \(\frac{19}{11}< \frac{19}{10}\)
Vậy \(S>1\frac{8}{11}\)
\(S=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=\frac{7-\left(b+c\right)}{b+c}+\frac{7-\left(c+a\right)}{c+a}+\frac{7-\left(a+b\right)}{a+b}\)
\(=\frac{7}{b+c}-\frac{b+c}{b+c}+\frac{7}{c+a}-\frac{c+a}{c+a}+\frac{7}{a+b}-\frac{a+b}{a+b}\)
\(=\frac{7}{b+c}-1+\frac{7}{c+a}-1+\frac{7}{a+b}-1\)
\(=\frac{7}{b+c}+\frac{7}{c+a}+\frac{7}{a+b}-3\)
\(=7.\left(\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}\right)-3\) \(.Thay\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=\frac{7}{10}\)
\(\Rightarrow S=7.\frac{7}{10}-3=\frac{49}{10}-3=1\frac{9}{10}>1\frac{8}{11}\)
Vậy\(S>1\frac{8}{11}\)
Xét: \(\left(a+b+c\right)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)=7\cdot\frac{7}{10}=\frac{49}{10}\)
\(\Leftrightarrow\frac{a+b}{a+b}+\frac{c}{a+b}+\frac{a+c}{a+c}+\frac{b}{a+c}+\frac{b+c}{b+c}+\frac{a}{b+c}=\frac{49}{10}\)
\(\Rightarrow3+\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=\frac{49}{10}\Rightarrow A=\frac{19}{10}\)
Vì \(19=19\) và \(\frac{1}{11}< \frac{1}{10}\Rightarrow\frac{19}{11}< \frac{19}{10}\Rightarrow S>\frac{19}{11}\)
Có 20/39>1/2; 18/41<1/2 suy ra 20/39>18/41
22/27>22/29
18/43 = 1- 25/43
14/39 = 1- 25/ 39
mà 25/43< 25/43 suy ra 18/43> 14/39 (vì cùng 1 số mà trừ đi số nhỏ hơn thì sẽ lớn hơn số đó mà lại đem trừ đi số lớn hơn)
Vậy A>B