Question

Tìm số tự nhiên n, biết :

2^2n-1 - 2 = 2 + 2² + 2³ + ...... + 2⁹⁹ + 2¹⁰⁰

Answer 1

ĐặtA = \(2+2^2+2^3+...+2^{100}\)

\(\Rightarrow2A=2^2+2^3+2^4+...+2^{101}\)

\(\Rightarrow2A-A=\left(2^2+2^3+2^4+...+2^{101}\right)-\left(2+2^2+2^3+...+2^{100}\right)\)

\(\Rightarrow A=2^{101}-2=2^{2n-1}-2\)

\(\Rightarrow2^{2n-1}=2^{101}\Rightarrow2n-1=101\)

\(\Rightarrow n=51\)

Answer 2

Đặt \(A=2+2^2+2^3+...+2^{100}\)

\(2A=2.\left(2+2^2+...+2^{100}\right)\)

\(2A=2^2+2^3+...+2^{101}\)

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

\(A=2^{101}-2\)

Ta có : \(2^{2n-1}-2=2^{101}-2\)

\(\Rightarrow2^{2n-1}=2^{101}\)

\(\Rightarrow2n-1=101\)

\(\Rightarrow n=51\)