Question

cho S=1+2+2^2+2^3+...+2^9

hãy so sánh S với 5 nhân 2^8

Answer 1

\(S=1+2+2^2+...+2^9\)

\(\Rightarrow2S=2+2^2+2^3+...+2^{10}\)

\(\Rightarrow2S-S=\left(2+2^2+2^3+...+2^{10}\right)-\left(1+2+2^2+...+2^9\right)\)

\(\Rightarrow S=2^{10}-1< 2^{10}=2^7.2^3=2^7.8\)

Do \(5.2^8=5.2.2^7=10.2^7>2^7.8\) nên \(5.2^8>2^{10}>2^{10}-1\)

\(\Rightarrow5.2^8>2^{10}-1\)

Vậy \(5.2^8>2^{10}-1\)

Answer 2

S = 1 + 2 + 2² + 2³ + ... + 2⁹

2S = 2 + 2² + 2³ + 2⁴ + ... + 2¹⁰

2S - S = (2 + 2² + 2³ + 2⁴ + ... + 2¹⁰) - (1 + 2 + 2² + 2³ + ... + 2⁹)

S = 2¹⁰ - 1 < 2¹⁰ = 2².2⁸ = 4.2⁸ < 5.2⁸

=> S < 5.2⁸