Question

Cho \(A=3^1+3^2+3^3+...+3^{2006}\)

a, Thu gọn A

b, Tìm x để 2A+ 3= \(3^x\)

 

Answer 1

\(A=3+3^2+3^3+.......+3^{2006}\)

\(\Leftrightarrow3A=3^2+3^3+......+3^{2007}\)

\(\Leftrightarrow3A-A=3^{2007}-3\)

\(\Leftrightarrow2A=3^{2007}-3\)

\(\Leftrightarrow A=\frac{3^{2007}-3}{2}\)

\(\Leftrightarrow2A+3=2^{2007}\)

\(\Leftrightarrow2^{2007}=2^x\)

\(\Leftrightarrow x=2007\)

Answer 2

\(3A=3^2+3^3+....+3^{2007}\)

\(3A-A=\left(3^2+3^3+...+3^{2007}\right)-\left(3+3^2+...+3^{2006}\right)\)

\(2A=3^{2007}-3\)

\(A=\frac{3^{2007}-3}{2}\)

b)\(2A+3=3^x\)

\(2A=3^x-3\)

Mà:\(2A=3^{2007}-3\)

\(\Rightarrow x=2007\)