Question

Aratati ca:
1/10<1/90+1/91+...+1/99<1/9

Answer 1

O sa notez suma $\frac{1}{90} + \frac{1}{91}+...+ \frac{1}{99}$ cu S.

Observam ca:
$\frac{1}{90}= \frac{1}{90} \\ \frac{1}{91}< \frac{1}{90} \\ \frac{1}{92}< \frac{1}{90} \\ ............ \\ \frac{1}{99}< \frac{1}{90}$.

Insumand toate aceste relatii, obtinem $S<10* \frac{1}{90}= \frac{1}{9} .$

Observam ca:
$\frac{1}{90}> \frac{1}{100} \\ \frac{1}{91}> \frac{1}{100} \\ \frac{1}{92}> \frac{1}{100} \\ .............. \\ \frac{1}{99} > \frac{1}{100}$.

Insumand toate aceste relatii, obtinem $S>10* \frac{1}{100}= \frac{1}{10}.$

Deci $\frac{1}{10}< S< \frac{1}{9} .$