Use mathematical induction to prove the statement is true for all positive integers n.5 + + + . . . + = 6

Question

What will be an ideal response?

Answer 1

Answers may vary. Possible answer:
First we show that the statement is true when n = 1.
 For n = 1, we get 5 = 6
 Since 6 = 6 = 5 , P₁ 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,
P_k: 5 +  +  + . . . +  = 6  is assumed true.
On the basis of the assumption that P_k is true, we need to show that P_k+1 is true. 
P_k+1: 5 +  +  + . . . +  = 6
So we assume that  is true and add the next term,  to both sides of the equation.  
5 +  +  + . . . +  +   = 6 + 
5 +  +  + . . . +  +  = 6 -  + 

5 +  +  + . . . +  +  = 6 - 
5 +  +  + . . . +  +  = 6

 The last equation says that P_k+1 is true if P_k is assumed to be true. Therefore, by the principle of mathematical induction, the statement    is true for all natural numbers n.