Question

Chưng minh : 

1,  A= 1/2.4 + 1/4.6 + ... + 1/60.62 < 1/4 

2, B= 1/1.6 + 1/6.11 + ....+ 1/201. 206 < 1/5 

 

Lưu ý : / có nghĩa là phân số 

Ai nhanh mới sẽ tick cho

Answer 1

Chỉ trong hôm nay và sáng mai thôi nhé. Ok

Answer 2

Xét công thức tổng quát:

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

\(2A=\frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{60.62}\)

\(2A=\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{60}-\frac{1}{62}\)

\(2A=\frac{1}{2}-\frac{1}{62}\)

\(2A=\frac{30}{62}\)

\(\Rightarrow A=\frac{30}{124}=\frac{15}{62}< \frac{15}{60}=\frac{1}{4}\)

Tương tự với câu 2 tính 6B 

Answer 3

Ta có:

\(A=\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{60.62}\)

\(\Rightarrow A=\frac{1}{2}.\left(\frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{60.62}\right)\)

\(\Rightarrow A=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{60}-\frac{1}{62}\right)\)

\(\Rightarrow A=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{62}\right)=\frac{1}{2}.\frac{15}{31}=\frac{15}{62}\)

\(\Leftrightarrow\frac{15}{62}< \frac{1}{4}\Leftrightarrow A< \frac{1}{4}\left(dpcm\right)\)

\(B=\frac{1}{1.6}+\frac{1}{6.11}+...+\frac{1}{201.206}\)

\(\Rightarrow B=\frac{1}{5}.\left(\frac{5}{1.6}+\frac{5}{6.11}+...+\frac{5}{201.206}\right)\)

\(\Rightarrow B=\frac{1}{5}.\left(\frac{1}{1}-\frac{1}{6}+\frac{1}{6}-\frac{1}{11}+...+\frac{1}{201}-\frac{1}{206}\right)\)

\(\Rightarrow B=\frac{1}{5}.\left(\frac{1}{1}-\frac{1}{206}\right)=\frac{1}{5}.\frac{205}{206}=\frac{41}{206}\)

\(\Leftrightarrow\frac{41}{206}< \frac{1}{5}\Leftrightarrow B< \frac{1}{5}\left(dpcm\right)\)

Answer 4

1/2×4+1/4×6+...+1/60×62 =1/2×2×(1/2×4+1/4×6+...+1/60×62) =1/2×(2/2×4+2/4+6+...+2/60×62) =1/2×(1/2-1/4+1/4-1/6+...+1/60-1/62) =1/2×(1/2-1/62) =1/2×(31/62-1/62) =1/2×30/62 =1/2×15/31 =15/62