Question

Đố ai giải ra được bài này trong 5 phút nè:

Chứng minh:1/3+1/7+1/13+1/21+...+1/73+1/97<1

Answer 1

1/3 + 1/7 + 1/13 + 1/21 + ... + 1/73 + 1/97 < 1/2 + 1/6 + 1/12 + 1/20 + ... + 1/72 + 1/90

<=> 1/3 + 1/7 + 1/13 + 1/21 + ... + 1/73 + 1/97 < 1/1*2 + 1/2*3 + 1/3*4 + 1/4*5 + ... + 1/8*9 + 1/9*10

<=> 1/3 + 1/7 + 1/13 + 1/21 + ... + 1/73 + 1/97 < 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ... + 1/8 - 1/9 + 1/9 - 1/10

<=> 1/3 + 1/7 + 1/13 + 1/21 + ... + 1/73 + 1/97 < 1 - 1/10

<=> 1/3 + 1/7 + 1/13 + 1/21 + ... + 1/73 + 1/97 < 9/10 < 1

Vậy 1/3 + 1/7 + 1/13 + 1/21 + ... + 1/73 + 1/97 < 1

Answer 2

Sửa lại đề : Chứng minh \(A=\frac{1}{3}+\frac{1}{7}+\frac{1}{13}+....+\frac{1}{73}+\frac{1}{91}< 1\)

Ta có :

\(\frac{1}{3}=\frac{1}{1.2+1}< \frac{1}{1.2}\)

\(\frac{1}{7}=\frac{1}{2.3+1}< \frac{1}{2.3}\)

\(\frac{1}{13}=\frac{1}{3.4+1}< \frac{1}{3.4}\)

\(.....\)

\(\frac{1}{91}=\frac{1}{9.10+1}< \frac{1}{9.10}\)

\(\Rightarrow A< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{9.10}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{9}-\frac{1}{10}=1-\frac{1}{10}< 1\)(đpcm)