Question

So sánh

a) 2^700 va 5^300

b)  so sánh S =1 +2+2^2+2^3+....+2^50  với 2^51

Answer 1

\(a,2^{700}=\left(2^7\right)^{100}=128^{100}\)

\(5^{300}=\left(5^3\right)^{100}=125^{100}\)

Có \(128^{100}>125^{100}\Rightarrow2^{700}>5^{300}\)

\(b,S=1+2+2^2+...+2^{50}\)

\(\Rightarrow2S=2+2^2+2^3+...+2^{51}\)

\(\Rightarrow2S-S=S=2^{51}-1< 2^{51}\)

Answer 2

a) Ta có :

\(2^{700}=\left(2^7\right)^{100}=128^{100}\)

\(5^{300}=\left(5^3\right)^{100}=125^{100}\)

Vì \(128^{100}>125^{100}\)\(\Rightarrow\)\(2^{700}>5^{300}\)

Vậy  \(2^{700}>5^{300}\)

b) \(S=1+2+2^2+...+2^{50}\)

\(\Rightarrow2S=2+2^2+2^3+...+2^{51}\)

\(\Rightarrow2S-S=\left(2+2^2+2^3+...+2^{51}\right)-\left(1+2+2^2+...+2^{50}\right)\)

\(\Rightarrow S=2^{51}-1< 2^{51}\)

Vậy S < 2⁵¹

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