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Có:LCM(3,5,7)= 105
=>\(\frac{3x-5y}{2}\)=\(\frac{7y-3z}{3}\)=\(\frac{5z-7x}{4}\)sẽ bằng \(\frac{21\left(3x-5y\right)}{2.21}\)=\(\frac{15\left(7y-3z\right)}{3.15}\)=\(\frac{9\left(5z-7x\right)}{4.9}\)
Và bằng \(\frac{63x-105y}{42}\)=\(\frac{105y-45z}{45}\)=\(\frac{45z-63x}{36}\)
Áp dụng dãy tỉ số bằng nhau ta có:
\(\frac{63x-105y+105y-45z+45z-63x}{45+42+36}\)=0
=>3x-5y=0 ;7y-3z=0 ;5z-7x=0
Xét 3x-5y=0 và 7y-3z=0
Có: 3x=5y :7y=3z
=>\(\frac{x}{5}\)=\(\frac{y}{3}\);\(\frac{y}{3}\)=\(\frac{z}{7}\)
=>\(\frac{x}{5}\)=\(\frac{y}{3}\)=\(\frac{z}{7}\)
Áp dung dãy tỉ số bằng nhau ta có:
\(\frac{x+y+z}{5+3+7}\)=\(\frac{17}{15}\)
Do đó: \(\frac{x}{5}\)=\(\frac{17}{15}\)=>x=\(\frac{17}{3}\)
\(\frac{y}{3}\)=\(\frac{17}{15}\)=>y=\(\frac{17}{5}\)
\(\frac{z}{7}\)=\(\frac{17}{15}\)=>z=\(\frac{119}{15}\)
2.Thấy $15;117y$ chia hết cho 3
\Rightarrow $38x$ chia hết cho 3
\Rightarrow $x$ chia hết cho 3
Đặt $x=3a$ (a thuộc Z)
\Rightarrow PT trở thành: $38a+39y=5$
\Leftrightarrow $y=\dfrac{5-38a}{39}=\dfrac{a+5}{39}-a$
Đặt $ dfrac{a+5}{39} = b$ (b thuộc Z)
\Rightarrow $a=39b-5$
\Rightarrow $y=b- (39b-5)=5-38b$
$x=3 (39b-5)=...$
Với b nguyên
Nghiệm tổng quát: $(x;y)=(...;.....)$ với b nguyên
ta có:
1/1.2+1/3.4+1/5.6+...+1/49.50
=>1-1/2+1/3-1/4+1/5-1/6+...+1/49-1/50
=>(1+1/3+1/5+1/7+...+1/49)-(1/2+1/4+1/6+...+1/50)
=>(1+1/2+1/3+...+1/49+1/50)-(1/2+1/4+1/6+...+1/50).2
=>(1+1/2+1/3+...+1/49+1/50) -( 1+1/2+1/3+...+1/25)
=>1/26+1/27+1/28+...+1/50=1/26+1/27+1/28+...+1/50
hay 1/1.2+1/3.4+1/5.6+...+1/49.50=1/26+1/27+1/28+...+1/50
Trả lời:
1, Ta có: \(x+y=\frac{1}{2};y+z=\frac{1}{3};z+x=\frac{1}{4}\)
\(\Rightarrow x+y+y+z+z+x=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}\)
\(\Rightarrow2x+2y+2z=\frac{13}{12}\)
\(\Rightarrow2\left(x+y+z\right)=\frac{13}{12}\)
\(\Rightarrow x+y+z=\frac{13}{24}\)
\(\Rightarrow\hept{\begin{cases}x=\frac{13}{24}-\frac{1}{3}=\frac{5}{24}\\y=\frac{13}{24}-\frac{1}{4}=\frac{7}{24}\\z=\frac{13}{24}-\frac{1}{2}=\frac{1}{24}\end{cases}}\)
2, Ta có: \(x:y:z=3:5:\left(-2\right)\Rightarrow\frac{x}{3}=\frac{y}{5}=\frac{z}{-2}\)
Áp dụng tc dãy tỉ số bằng nhau, ta có:
\(\frac{x}{3}=\frac{y}{5}=\frac{z}{-2}=\frac{5x-y+3z}{5.3-5+3.\left(-2\right)}=\frac{124}{4}=31\)
\(\Rightarrow\hept{\begin{cases}x=93\\y=155\\z=-62\end{cases}}\)
3, Ta có: \(2x=3y\Rightarrow\frac{x}{3}=\frac{y}{2}\Rightarrow\frac{x}{21}=\frac{y}{14}\left(1\right)\)
\(5y=7z\Rightarrow\frac{y}{7}=\frac{z}{5}\Rightarrow\frac{y}{14}=\frac{z}{10}\left(2\right)\)
Từ (1) và (2) => \(\frac{x}{21}=\frac{y}{14}=\frac{z}{10}\)
Áp dụng tc dãy tỉ số bằng nhau, ta có:
\(\frac{x}{21}=\frac{y}{14}=\frac{z}{10}=\frac{3x-7y+5x}{3.21-7.14+5.10}=\frac{30}{15}=2\)
\(\Rightarrow\hept{\begin{cases}x=42\\y=28\\z=20\end{cases}}\)
a)A = 1 / (1*2) + 1 / (3*4) + ... + 1 / (99*100) > 1 / (1*2) + 1 / (3*4) = 1 / 2 + 1 / 12 = 7 / 12 ♦
A = 1 / (1*2) + 1 / (3*4) + ... + 1 / (99*100) = (1 - 1 / 2) + (1 / 3 - 1 / 4) + ... + (1 / 99 - 100) =
(1 - 1 / 2 + 1 / 3) - (1 / 4 - 1 / 5) - (1 / 6 - 1 / 7) - ... - (1 / 98 - 1 / 99) - 1 / 100 <
1 - 1 / 2 + 1 / 3 = 5 / 6 ♥
♦, ♥ => 7 / 12 < A < 5 / 6
b)ta có:
1/1.2+1/3.4+1/5.6+...+1/49.50
=>1-1/2+1/3-1/4+1/5-1/6+...+1/49-1/50
=>(1+1/3+1/5+1/7+...+1/49)-(1/2+1/4+1/6+...+1/50)
=>(1+1/2+1/3+...+1/49+1/50)-(1/2+1/4+1/6+...+1/50).2
=>(1+1/2+1/3+...+1/49+1/50) -( 1+1/2+1/3+...+1/25)
=>1/26+1/27+1/28+...+1/50=1/26+1/27+1/28+...+1/50
hay 1/1.2+1/3.4+1/5.6+...+1/49.50=1/26+1/27+1/28+...+1/50
\(2\)
CMR
\(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+....+\frac{1}{49.50}=\frac{1}{26}+\frac{1}{27}+\frac{1}{28}+...+\frac{1}{50}\)
\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)
\(=\left(\frac{1}{1}+\frac{1}{3}+...+\frac{1}{49}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{50}\right)\)
\(=\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{49}+\frac{1}{50}\right)-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{50}\right)\)
\(=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{50}-1-\frac{1}{2}-\frac{1}{3}-...-\frac{1}{25}\)
\(=\frac{1}{26}+\frac{1}{27}+....+\frac{1}{50}\left(đpcm\right)\)
mình biết làm mỗi ý thứ 2 thôi.