Question

Cho biểu thức M = \(\frac{1}{5}\)+ \(\left(\frac{1}{5}\right)^2\)+ \(\left(\frac{1}{5}\right)^3\)+...+ \(\left(\frac{1}{5}\right)^{50}\). Chứng minh M < 0,25

Answer 1

\(M=\frac{1}{5}+\left(\frac{1}{5}\right)^2+...+\left(\frac{1}{5}\right)^{50}\)

\(M=\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{50}}\)

\(5M=5\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{50}}\right)\)

\(5M=1+\frac{1}{5}+...+\frac{1}{5^{49}}\)

\(5M-M=\left(1+\frac{1}{5}+...+\frac{1}{5^{49}}\right)-\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{50}}\right)\)

\(4M=1-\frac{1}{5^{50}}\)

\(M=\frac{1-\frac{1}{5^{50}}}{4}< \frac{1}{4}=0,25\)

Đpcm