ongelijkheid

Opgave - VWO 2002 vraag 3

Toon aan dat $$\frac1{15}<\frac12\cdot\frac34\cdot\frac56\cdot...\cdot\frac{97} {98}\cdot\frac{99}{100}<\frac1{10}.$$

Oplossing

Zij $A=\frac12\cdot\frac34\cdots\frac{99}{100}$ en $B=\frac23\cdot\frac45\cdots\frac{98}{99}$, zodat $AB=\frac{1}{100}$.

  • Enerzijds is $\frac23<\frac34, \frac45<\frac56, \cdots , \frac{98}{99}<\frac{99}{100}$ dus $2A>B$.
  • Anderzijds is $\frac12<\frac23,\frac34<\frac45, \cdots, \frac{99}{100}<1$ dus $A

Dus is $\frac{1}{200}