Question

Tính giá trị biểu thức: M=2/3+2/9+2/27+2/81+...+2/729

Answer 1

Đặt \(N=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^6}\)

=>\(3N=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^5}\)

=>\(3N-N=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^5}-\frac{1}{3}-\frac{1}{3^2}-\frac{1}{3^3}-...-\frac{1}{3^6}\)

=>\(2N=1-\frac{1}{3^6}\)

=>\(2N=1-\frac{1}{729}=\frac{729}{729}\)

Lại có:\(M=\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+...+\frac{2}{729}\)

=>\(M=2.\left(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{729}\right)\)

=>\(M=2.\left(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^6}\right)\)

=>\(M=2.N\)

=>\(M=\frac{728}{729}\)