# What's the Final Number ?

Number Theory Level 2

Consider the set

$S= \left \{ 1, \frac {1}{2}, \frac {1}{3}, \frac {1}{4},\cdots, \frac {1}{100} \right \}.$

Choose any two numbers $$x$$ and $$y,$$ and replace them with $$x+y+ xy.$$

For example, if we choose the numbers $$\frac{1}{2}$$ and $$\frac {1}{8}$$, we will replace them by $$\frac {11}{16}$$.

If we keep repeating this process until only $$1$$ number remains, what is the final number?

×