# What's the Final Number ?

**Number Theory**Level 2

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

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?