Question

Chứng minh rằng :

\(\dfrac{1}{7^2} - \dfrac{1}{7^4} + ... + \dfrac{1}{7^{4n-2}} - \dfrac{1}{7^{4n}} + ... + \dfrac{1}{7^{98}}-\dfrac{1}{7^{100}} < \dfrac{1}{50}\)

Answer 1

\(\text{Đặt:}S=\frac{1}{7^2}-\frac{1}{7^4}+....-\frac{1}{7^{100}}\Rightarrow49S=1-\frac{1}{7^2}+.....-\frac{1}{7^{98}}\Rightarrow49S+S=50S=\left(1-\frac{1}{7^2}+\frac{1}{7^4}-....-\frac{1}{7^{98}}\right)+\left(\frac{1}{7^2}-\frac{1}{7^4}+....-\frac{1}{7^{100}}\right)=1-\frac{1}{7^{100}}< 1\Rightarrow S< \frac{1}{50}\left(\text{đpcm}\right)\)