Provide an appropriate response.Prove by induction that the number of diagonals of a polygon with n sides (n ? 4) is .

Question

What will be an ideal response?

Answer 1

Answers may vary. Possible answer:
First, we show the statement is true when n = 4.
When n = 4, the polygon is a rectangle which has 2 diagonals. When n = 4,  =  = 2
So P₄ is true and the first condition for the principle of induction is satisfied.
Next, we assume the statement holds for some unspecified natural number k. That is, we assume that P_k is true:
P_k: the number of diagonals of a polygon with k sides is  .
On the basis of the assumption that P_k is true, we need to show that P_k+1 is true. 
P_k+1: the number of diagonals of a polygon with k + 1 sides is  . 
Start with a polygon having k sides. To create a polygon with k + 1 sides, remove one of the existing sides and create 2 sides where there was one. So an additional vertex is created. The number of diagonals of the new polygon is the number of diagonals of the original k-sided polygon plus the number of new diagonals that can be drawn from the (k + 1)st vertex (= k + 1 - 3) plus the additional diagonal that can be drawn in place of the side which was removed.  
So the number of diagonals of the new polygon =  + (k + 1 - 3) + 1 
 =   + (k - 1)
 =  
 = 
 = 

So P_k+1 is true if P_k is assumed true. Therefore, by the principle of mathematical induction, the number of diagonals of a polygon with n sides (n ? 4) is  .