Cho `A = 1/5 + 1/5^2 + 1/5^3 +...+ 1/5^2016`. Chứng minh `A < 1/4`

Ta có:

`A = 1/5 + 1/5^2 + 1/5^3 + ... + 1/5^2016`

`5A = 5(1/5+1/5^2+1/5^3+...+1/5^2016)`

`5A = 1 + 1/5 + 1/5^2 + ... + 1/5^2015`

`5A - A = (1+1/5+1/5^2+...+1/5^2015) - (1/5+1/5^2+1/5^3+...+1/5^2016)`

`4A = 1 - 1/5^2016`

`A = (1-1/5^2016)/4`

Vì `(1-1/5^2016)/4 < 1/4`

`=> A < 1/4` (đpcm)