Chứng minh rằng:1/5 +1/15 +1/25 +…+1/1985<9/20

Giải thích các bước giải:

Ta có :
$A=\dfrac{1}{5}+\dfrac{1}{15}+\dfrac{1}{25}+..+\dfrac{1}{1985}$ 

$\to 5A=1+\dfrac{1}{3}+\dfrac{1}{5}+..+\dfrac{1}{397}$ 

$\to 5A=1+\dfrac{1}{3}+(\dfrac{1}{5}+\dfrac{1}{7}+\dfrac{1}{9})+(\dfrac{1}{11}+\dfrac{1}{13}+..+\dfrac{1}{27})+(\dfrac{1}{29}+\dfrac{1}{31}+..+\dfrac{1}{81})+(\dfrac{1}{83}+\dfrac{1}{85}+..+\dfrac{1}{243})+..+\dfrac{1}{397}$ 

$\to 5A\ge 1+\dfrac{1}{3}+3.\dfrac{1}{9}+9.\dfrac{1}{27}+27.\dfrac{1}{81})+81.\dfrac{1}{243}$ 

$\to 5A\ge \dfrac 94$

$\to A\ge \dfrac{9}{20}$