Chapter 8.5, Problem 66PE

Describe how you would use mathematical induction to prove

${(a+b)}^n=\left(\begin{array}{l}n\\ 0\end{array}\right)a^n+\left(\begin{array}{l}n\\ 1\end{array}\right)a^{n-1}b+\left(\begin{array}{l}n\\ 2\end{array}\right)a^{n-2}{b}^2+\cdots +\left(\begin{array}{l}n\\ n-1\end{array}\right)a{b}^{n-1}+\left(\begin{array}{l}n\\ n\end{array}\right)b^n.$

What happens when n - 1? Write the statement that we assume to be true. Write the statement that we must prove. What must be done to the left side of the assumed statement to make it look like the left side of the statement that must be proved? (More detail on the actual proof is found In Exercise 85.)