Chapter 6.2, Problem 9MC

a. With the exception of $\frac{2}{3}$, the Egyptians used only unit fractions(fractions that have numerators of $1$). Every unit fraction can be expressed as the sum of two unit fractions in more than one way, for example, $\frac{1}{2}=\frac{1}{4}+\frac{1}{4}$ and $\frac{1}{2}=\frac{1}{3}+\frac{1}{6}$. Find at least two different unit fraction representations for each of the following.

i. $\frac{1}{3}$

ii. $\frac{1}{7}$

b. Show that $\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n\left(n+1\right)}$.

c. Rewrite the equation in part (b) as a sum and then use the sum to answer part(a).

d. Write $\frac{1}{17}$ as a sum of two different unit fractions.