Question

So sánh:

A = 20¹⁹+1 / 20²⁰+1 và B = 20²⁰+1 / 20²¹+1   

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Answer 1

Ta có:

\(20A=\frac{20\left(20^{19}+1\right)}{20^{20}+1}=\frac{20^{20}+20}{20^{20}+1}=\frac{20^{20}+1+19}{20^{20}+1}=\frac{20^{20}+1}{20^{20}+1}+\frac{19}{20^{20}+1}=1+\frac{19}{20^{20}+1}\)

\(20B=\frac{20\left(20^{20}+1\right)}{20^{21}+1}=\frac{20^{21}+20}{20^{21}+1}=\frac{20^{21}+1+19}{20^{21}+1}=\frac{20^{21}+1}{20^{21}+1}+\frac{19}{20^{21}+1}=1+\frac{19}{20^{21}+1}\)

Vì 20²⁰+1<20²¹+1

\(\Rightarrow\frac{19}{20^{20}+1}>\frac{19}{20^{21}+1}\)

\(\Rightarrow1+\frac{19}{20^{20}+1}>1+\frac{19}{20^{21}+1}\)

\(\Rightarrow20A>20B\)

\(\Rightarrow A>B\)