Question

chứng minh 1+2+2²+...+2⁹⁹+2¹⁰⁰ = 2¹⁰¹-1

Answer 1

3²¹>2³¹

Answer 2

3^{21 _{> 2}31}

Answer 3

\(3^{21}=\left(3^3\right)^7=27^7\)

\(2^{31}=\left(2^4\right)^7.2=\left(2^5\right)^7=32^7\)

Vì 27 < 32 nên \(3^{21}< 2^{31}\)

Answer 4

3^2>2^3 -->(3^2)^10>(2^3)^10 --> 3^20>2^30
-->3^21>2^31
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Answer 5

Đặt \(A=1+2+2^2+...+2^{99}+2^{100}\Rightarrow2A=2+2^2+2^3+...+2^{101}\)

\(\Rightarrow2A-A=\left(2+2^2+2^3+...+2^{101}\right)-\left(1+2+2^2+...+2^{100}\right)\)

\(A=2^{101}-1\)=> đpcm