Question

Cho B= \(\frac{1}{2}+\left(\frac{1}{2}\right)^2+....+\left(\frac{1}{2}\right)^{2014}+\left(\frac{1}{2}\right)^{2015}\).Chứng minh B<1

Answer 1

\(B=\frac{1}{2}+\left(\frac{1}{2}\right)^2+...+\left(\frac{1}{2}\right)^{2015}\)

\(B=\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2015}}\)

\(2B=2\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2015}}\right)\)

\(2B=1+\frac{1}{2}+...+\frac{1}{2^{2014}}\)

\(2B-B=\left(1+\frac{1}{2}+...+\frac{1}{2^{2014}}\right)-\left(\frac{1}{2}+\frac{1}{2^2}+...+\frac{1}{2^{2015}}\right)\)

\(B=1-\frac{1}{2^{2015}}< 1\). Vậy ta có điều phải chứng minh