Question

Só sánh 

Cho A = 3 +2^2+2^3+2^4+...+2^2001 và B = 2^2003

Cho C = 4 +3^2+3^3+3^4+...3^2003+3^2004 và D = 3^2005

Answer 1

\(A=3+2^2+2^3+2^4+..+2^{2001}\)

\(\Rightarrow A=1+2+2^2+2^3+2^4+...+2^{2001}\)

\(\Rightarrow2A=2+2^2+2^3+...+2^{2002}\)

\(\Rightarrow2A-A=\left(2+2^2+2^3+...+2^{2002}\right)-\left(1+2+3^2+...+2^{2001}\right)\)

\(\Rightarrow A=2^{2002}-1\)

Vì \(2^{2002}-1< 2^{2003}\) nên \(A< 2^{2003}\)

Answer 2

Ta có:

\(C=4+3^2+3^3+...+3^{2003}+3^{2004}\)

\(C=1+3+3^2+3^3+...+3^{2003}+3^{2004}\)

\(\Rightarrow3C=3+3^2+3^3+...+3^{2004}+3^{2005}\)

\(\Rightarrow3C-C=\left(3+3^2+3^2+...+3^{2004}+3^{2005}\right)-\left(1+3+3^2+3^3+...+3^{2003}+3^{2004}\right)\)

\(\Rightarrow2C=3^{2005}-1\)

\(\Rightarrow C=\left(3^{2005}-1\right):2< 3^{2005}\)

\(\Rightarrow C< 3^{2005}\)