Question

Cho A = 1 + 4 + 4² + ... + 4⁹⁹ và B = 4¹⁰¹

Chứng minh rằng: A/B  < 1/3  

Answer 1

\(4A=4+4^2+4^3+...+4^{100}\)

\(\Rightarrow3A=4A-A=4^{100}-1\Rightarrow A=\frac{4^{100}-1}{3}\)

Do đó \(\frac{A}{b}=\frac{\frac{4^{100}-1}{3}}{4^{101}}=\frac{4^{100}-1}{4^{101}.3}< \frac{4^{101}}{4^{101}.3}=\frac{1}{3}\)

Answer 2

Ta có : \(A=1+4+4^2+4^3+...+4^{99}\)

\(4A=4+4^2+4^3+4^4+...+4^{100}\)

\(4A-A=4^{100}-1\)

\(3A=4^{100}-1\)

\(A=4^{100}-1\div3\)

ta thấy : \(4^{100}-1\div3< 4^{101}\)

\(\Rightarrow A< B\)