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

Question

What will be an ideal response?

Answer 1

Step 1: List terms of series.
  S_n: 12 + 22 + 32 + . . . + n² =  
Step 2:  Determine if S₁, S₂, and S₃ are true. 
      
  
    
Step 3: Assume S_k to be true, then
   S_k: 12 + 22 + 32 + . . . + k² = 
Step 4: Find S_k+1.  
   S_k+1: 12 + 22 + 32 + . . . + (k + 1)² = 
  S_k+1: 12 + 22 + 32 + . . . + (k + 1)² = 
Step 5: Truth of S_k implies that S_k+1 is true.  
   12 + 22 + 32 + . . . + k² = 
  12 + 22 + 32 + . . . + k² + (k + 1)² =  + (k + 1)2
  12 + 22 + 32 + . . . + k² + (k + 1)² =  + (k + 1)(k + 1)
  12 + 22 + 32 + . . . + k² + (k + 1)² =  + 
  12 + 22 + 32 + . . . + k² + (k + 1)² =   
  12 + 22 + 32 + . . . + k² + (k + 1)² =   
  12 + 22 + 32 + . . . + k² + (k + 1)² =   
  12 + 22 + 32 + . . . + k² + (k + 1)² =   
Since the last equation is S_k+1, it has been shown that the truth of S_k implies the truth of S_k+1 for every positive integer k. By the principle of mathematical induction, S_n is true for every positive integer n.