A=1/1.2+1/3.4+1/5.6+....+1/97.98+1/99.100 B=1/50+1/51+1/52+....+1/99+1/100 Tính A-B
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\(A=\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{99.100}\)
\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}\)
\(=1-\frac{1}{100}\)
\(=\frac{99}{100}\)
Nhầm tưởng tính tích :v
Ta có :
\(B=\frac{1}{50}+\frac{1}{51}+...+\frac{1}{99}+\frac{1}{100}< \frac{1}{51}+\frac{1}{51}+...+\frac{1}{51}=50.\frac{1}{51}=\frac{50}{51}< \frac{99}{100}\)
\(\Leftrightarrow A>B\)
xét B ta có:
B=1/1.2+1/3.4+1/5.6+...+1/99.100
B=1-1/2+1/3-1/4+1/5-1/6+...+1/99-100
B=(1+1/3+1/5+...+1/99)-(1/2+1/4+...+1/100)
B=(1+1/3+1/5+...+1/99)+(1/2+1/4+1/6+...+1/100)-2(1/2+1/4+1/6+...+1/100)
B=(1+1/2+1/3+...+1/99+1/100)-(1+1/2+1/3+1/4+...+1/50)
=>B=1/51+1/52+1/53+...+1/100
=>A/B=1/51+1/52+...+1/100:1/51+1/52+...+1/100=1 (đpcm)
Đó là cách nhanh nhất để giải nếu bn ko hỉu thì mik sẽ giải chi tiết cho
chúc bn học tốt ^-^
B=1/1.2+1/3.4+1/5.6+...+1/99.100
=1-1/2+1/3-1/4+1/5-1/6+...+1/99-1/100
=(1+1/3+1/5+...+1/99)-(1/2+1/4+1/6+...+1/100)
=(1+1/2+1/3+1/4+1/5+1/6+...+1/99+1/100)-2(1/2+1/4+1/6+...+1/100)
=(1+1/2+1/3+1/4+...+1/100)-(1+1/2+1/3+..+1/50)
=1/51+1/52+1/53+..+1/100 (1)
A=1/51+1/52+1/53+..+1/100 (2)
(1),(2)=> A/B=1
Ta có : B=1-1/2+1/3-...+1/99-1/100= ( 1+1/3+...+1/99) -(1/2+....+1/100)= ( 1+1/2+1/3+....+1/99+1/100)-2.(1/2+...+1/100) =1+1/2+1/3+...+1/100 - ( 1+...+1/50) = (1+1/2+...+1/50) + ( 1/51+1/52+...+1/100) - ( 1+...+1/50)= 1/51 +1/52+...+1/100 (1)
C=1/(1.2) +1/(3.4) +...+1/(99.100) = 1-1/2+ 1/3-1/4+...+1/99-1/100 =...
Biểu thức C phần còn lại làm tương tự giống phần (1) nhé => C= 1/51+1/52+...+1/100 (2)
A=1/51+...+1/100(3)
Từ (1),(2) và (3)=>A=B=C (đpcm) . Chúc cậu học tốt !
Answer:
Mình làm thành tính tỉ số luôn nhé!
\(A=\frac{\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}}{\frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{99.100}}\)
Ta xét \(B=\frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{99.100}\)
\(=\frac{2-1}{1.2}+\frac{4-3}{3.4}+...+\frac{100-99}{99.100}\)
\(=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(=\left(1+\frac{1}{3}+...+\frac{1}{99}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)\)
\(=\left(1+\frac{1}{3}+...+\frac{1}{99}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)+\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)\)
\(=\left(1+\frac{1}{3}+...+\frac{1}{99}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)+\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{100}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}+\frac{1}{100}\right)-\left(1+\frac{1}{2}+...+\frac{1}{50}\right)\)
\(=1+\frac{1}{2}+...+\frac{1}{100}-1-\frac{1}{2}-...-\frac{1}{50}\)
\(=\left(1-1\right)+\left(\frac{1}{2}-\frac{1}{2}\right)+...+\left(\frac{1}{50}-\frac{1}{50}\right)+\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}\)
\(=\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}\)
\(\Rightarrow\frac{A}{B}=\frac{\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}}{\frac{1}{51}+\frac{1}{52}+...+\frac{1}{100}}\)
\(\Rightarrow\frac{A}{B}=1\)