Solution:

For $n=3$, we can write
$$1 = \frac{1}{2} + \frac{1}{3} + \frac{1}{6}.$$
For $n=4$, we can write
$$1 = \frac{1}{2} + \frac{1}{3} + \frac{1}{9} + \frac{1}{18}.$$
We claim that we can continue this pattern by breaking the smallest fraction into two smaller fractions, such that the smallest fraction always has even denominator. We can show this inductively. Assume this is true for some $n$, where the smallest fraction is $\frac{1}{2m}$. Then we can break the $\frac{1}{2m}$ into
$$\frac{1}{2m} = \frac{1}{3m} + \frac{1}{6m}$$
which creates a sum of $n+1$ distinct fractions with numerator $1$, and the largest denominator is still even. This completes our induction.