Chapter 12.4, Problem 32AYU

Extended Principle of Mathematical Induction The Extended Principle of Mathematical Induction states that if Conditions a and b hold, that is,

a. A statement is true for a natural number $j$.

b. If the statement is true for some natural number $k\ge j$, then it is also true for the next natural number $k+1$.

Then the statement is true for all natural numbers $\ge j$. Use the Extended Principle of Mathematical Induction to show that the number of diagonals in a convex polygon of $n$ sides is $\frac{1}{2}n\left(n-3\right)$.

[Hint: Begin by showing that the result is true when $n\text{ = }4$ (Condition (a).]