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Ta có: \(\frac{1}{x}+\frac{1}{y}=\frac{1}{p}\)⇔ p(x+y)=xy (1)
Vì p là số nguyên tố nên suy ra trong hai số x,y luôn có 1 số chia hết cho p.
Không mất tính tổng quát ta giả sử: x ⋮ p ⇒ x=kp (k∈N∗)
Nếu k=1, thay vào (1) ta được: p(p+y)=p ⇒ p+y=1, vô lí.
Do đó k≥2. Từ (1) suy ra: p(kp+y)=kp.y ⇔ y=\(\frac{kp}{k-1}\)
Do y∈N∗ mà (k;k−1)=1 ⇒ p ⋮ k−1 ⇒ k−1∈{1;p}
∙ k−1=1 ⇒ k=2⇒x=y=2p
∙ k−1 = p ⇒ k=p+1 ⇒ x=p(p+1),y=p+1
Vậy phương trình có ba nghiệm là: (2p;2p),(p+1;p2+p),(p2+p;p+1).
a) \(\frac{x}{4}=\frac{16}{x^2}\)\(=>x^3=16.4\)\(=>x^3=64\)\(=>x=4\)
b) \(\frac{4}{3}:\frac{4}{5}=\frac{2}{3}.\left(\frac{1}{10}.x\right)\)\(=>\frac{4}{3}.\frac{5}{4}=\frac{2}{3}\left(\frac{1}{10}x\right)\)\(=>\frac{5}{3}=\frac{2}{3}\left(\frac{1}{10}x\right)\)\(=>\frac{5}{3}:\frac{2}{3}=\frac{1}{10}x\)\(=>\frac{5}{3}.\frac{3}{2}=\frac{1}{10}x\)\(=>\frac{5}{2}=\frac{1}{10}x\)\(=>x=\frac{5}{2}:\frac{1}{10}\)\(=>x=\frac{5}{2}.10\)\(=>x=25\)
vậy x=25
1.
a) \(\frac{x}{4}=\frac{16}{x^2}\)
\(\Rightarrow x^3=64\)
\(\Rightarrow x^3=4^3\)
\(\Rightarrow x=4\)
b) \(1\frac{1}{3}:0,8=\frac{2}{3}.\left(0,1.x\right)\)
\(\frac{5}{3}=\frac{2}{3}.\frac{x}{10}\)
\(\frac{x}{10}=\frac{5}{2}\)
\(\Rightarrow x=\frac{5.10}{2}=25\)
2.
\(A=\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}+\frac{1}{3^{99}}\)
\(3A=1+\frac{1}{3}+...+\frac{1}{3^{97}}+\frac{1}{3^{98}}\)
\(3A-A=\left(1+\frac{1}{3}+...+\frac{1}{3^{97}}+\frac{1}{3^{98}}\right)-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}+\frac{1}{3^{99}}\right)\)
\(2A=1-\frac{1}{3^{99}}< 1\)
\(\Rightarrow A=\frac{1-\frac{1}{3^{99}}}{2}< \frac{1}{2}\)
1) Tính C
\(C=\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+....+\frac{n-1}{n!}\)
\(=\frac{2-1}{2!}+\frac{3-1}{3!}+\frac{4-1}{4!}+...+\frac{n-1}{n!}\)
\(=1-\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+...+\frac{1}{\left(n-1\right)!}-\frac{1}{n!}\)
\(=1-\frac{1}{n!}\)
3) a) Ta có : \(P=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{199}-\frac{1}{200}\)
\(=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{199}+\frac{1}{200}-2\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{200}\right)\)
\(=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{199}+\frac{1}{200}-1-\frac{1}{2}-\frac{1}{3}-...-\frac{1}{100}\)
\(=\frac{1}{101}+\frac{1}{102}+....+\frac{1}{199}+\frac{1}{200}\left(đpcm\right)\)
\(B=\left(1-\frac{3}{2.4}\right)\left(1-\frac{3}{3.5}\right)\left(1-\frac{3}{4.6}\right)...\left(1-\frac{3}{n\left(n+2\right)}\right)\)
\(=\frac{1.5}{2.4}.\frac{2.6}{3.5}.\frac{3.7}{4.6}...\frac{\left(n-1\right)\left(n+3\right)}{n\left(n+2\right)}\)
\(=\frac{\left[1.2.3...\left(n-1\right)\right]\left[5.6.7...\left(n+3\right)\right]}{\left(2.3.4...n\right)\left[4.5.6...\left(n+2\right)\right]}\)
\(=\frac{n+3}{4n}< 2\left(đpcm\right)\)