Question

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

a,thu gọn A

b, tìm x để 2A + 3= \(3^x\)

Answer 1

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

=> \(3A=3^2+3^3+......+3^{2007}\)

=> \(3A-A=3^{2007}-3\)

=> \(2A=3^{2007}-3\)

=> \(A=\frac{3^{2007}-3}{2}\)

b) Ta có : \(2A=3^{2007}-3\) (theo ý a)

=> \(2A+3=3^{2007}\)

=> x = 2007

Answer 2

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

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

\(\Leftrightarrow3A-A=\left(3^2+3^3+.......+3^{2007}\right)-\left(3+3^2+.....+3^{2006}\right)\)

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

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

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

\(\Leftrightarrow3^x=3^{2007}\)

\(\Leftrightarrow x=2007\left(tm\right)\)

Answer 3

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

=>\(3A=3\left(3+3^2+3^3+...+3^{2006}\right)=3^2+3^3+3^4+...+3^{2007}\)

=>\(3A-A=\left(3^2+3^3+3^4+...+3^{2007}\right)-\left(3+3^2+3^3+...+3^{2006}\right)\)

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

b)\(2A+3=3^x\Rightarrow2.\frac{3^{2007}-3}{2}+3=3^x\Rightarrow3^{2007}=3^x\Rightarrow x=2007\)