Question

[Hoàng Tử Tuấn Minh]

Bài 1 Tìm x,y thuộc số nguyên

5/x - y/3 =1/6

Bài 2 Chứng minh

A) A= 1/4^2 + 1/6^2 + .... + 1/(2^n)^2 < 1/4

B) B= 2!/3! + 2!/4! +...+2!/n! < 1

Bài 3 Cho

C = 1/41 + 1/42 + .... + 1/80

Chứng minh 7/12 < C < 5/6

Bài 4 Tìm n thuộc số nguyên biết :

A = 19/n-1 nhân n/9 sao cho thuộc số nguyên

Bài 5 Tính

A) 1/3 + 1/3^2 + 1/3^2 + .... + 1/3^100

B) 1/5 - 1/5^2 + 1/5^3 - 1/5^4 + ..... + 1/5^99 - 1/5^100

Answer 1

[Phương Trâm]

Bài 1:

\(\dfrac{5}{x} - \dfrac{y}{3} =\dfrac{1}{6}\)

\(\Rightarrow\dfrac{1}{6}+\dfrac{y}{3}=\dfrac{5}{x}\)

\(\Rightarrow\dfrac{1}{6}+\dfrac{2y}{6}=\dfrac{5}{x}\)

\(\Rightarrow1+\dfrac{2y}{6}=\dfrac{5}{x}\)

\(\Rightarrow x.\left(1+2y\right)=30\)

Vì \(2y\) chẵn nên \(1+2y\) lẻ

\(\Rightarrow1+2y\in\left\{\pm1;\pm3;\pm5;\pm30\right\}\)

\(\Rightarrow x\in\left\{\pm10;\pm30;\pm6;\pm2\right\}\)

Answer 2

[Phương Trâm]

Bài 2:

\(\dfrac{1}{4^2}+\dfrac{1}{6^2}+...+\dfrac{1}{\left(2n\right)^2}< \dfrac{1}{2.4}+\dfrac{1}{4.6}+\dfrac{1}{6.8}+...+\dfrac{1}{\left(2n-2\right).2n}\)

\(=\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{\left(2n-2\right).2n}\right).\dfrac{1}{2}\)

\(=\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{12}+...+\dfrac{1}{2n-2}-\dfrac{1}{2n}\right).\dfrac{1}{2}\)

\(=\left(\dfrac{1}{2}-\dfrac{1}{2n}\right).\dfrac{1}{2}\)

\(=\dfrac{1}{4}-\dfrac{1}{2n.2}< \dfrac{1}{4}\)

\(\Rightarrow\dfrac{1}{4^2}+\dfrac{1}{6^2}+\dfrac{1}{8^2}+...+\dfrac{1}{\left(2n\right)^2}< \dfrac{1}{4}\left(đpcm\right)\)