Question

a- Cho A= 1+2+2^2+2^3+...+2^9 và B= 5*2^8. Hãy so sánh A và B

b- So sánh: A= 2/60*63+2/63*66+...+2/117*120+2/2003

B= 5/40*44+5/44*48+...+5/76*80+5/2003

Answer 1

b: \(A=\dfrac{2}{3}\left(\dfrac{1}{60}-\dfrac{1}{63}+\dfrac{1}{63}-\dfrac{1}{66}+...+\dfrac{1}{117}-\dfrac{1}{120}\right)+\dfrac{2}{2003}\)

\(=\dfrac{2}{3}\cdot\dfrac{1}{120}+\dfrac{2}{2003}\)

\(=2\left(\dfrac{1}{360}+\dfrac{1}{2003}\right)\)

\(B=\dfrac{5}{4}\left(\dfrac{1}{40}-\dfrac{1}{44}+\dfrac{1}{44}-\dfrac{1}{48}+...+\dfrac{1}{76}-\dfrac{1}{80}\right)+\dfrac{5}{2003}\)

\(=\dfrac{5}{4}\cdot\dfrac{1}{80}+\dfrac{5}{2003}\)

\(=5\left(\dfrac{1}{320}+\dfrac{1}{2003}\right)\)

Vì 1/360+1/2003<1/320+1/2003

nên A<B