Question

[Nguyễn Đăng Khôi]

So sánh 1+2²+2³+...+2²⁰⁰⁵ với 2²⁰⁰⁶

Answer 1

[Tung Duong]

1 + 2² + 2³ + ... + 2²⁰⁰⁵

Gọi dãy số trên là A

A = \(1+2^2+2^3+....+2^{2005}\)

A =\(2^0+2^2+2^3+....+2^{2005}\)

A + \(2^1\)=  \(2^0+2^1+2^2+2^3+....+2^{2005}\)

( A + 2 ) x 2¹ = \(\left(2^0+2^1+2^2+2^3+....+2^{2005}\right)\times2^1\)

Ax2 + 4 =\(2^1+2^2+2^3+2^4+....+2^{2006}\)

4 + A x 2 - A =\(2^1+2^2+2^3+2^4+....+2^{2006}-\left(1+2^2+2^3+...2^{2005}\right)\)

4 + A = \(2^1+2^2+2^3+2^4+....+2^{2006}-1-2^2-2^3-....-2^{2005}\)

4 + A = \(2^{2006}-1\)

A=\(2^{2006}-1-4\)

A = \(2^{2006}-5\)

Mà \(2^{2006}-5< 2^{2006}\) 

\(\Rightarrow1+2^2+2^3+....+2^{2005}< 2^{2006}\)