Use mathematical induction to prove that the statement is true for every positive integer n.4 + 4 ? + 4 ? 2 + . . . + 4 ? n - 1 =

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. Then, 3 =  =   = 3. Thus, the statement is true for n = 1.
b). Assume that the statement is true for n = k:
 S_k = 
 Also, if the statement is true for n = k + 1, then
 S_k+1 = S_k + 4 ? ^{(k+1) - 1}= 
 Substitute the expression for S_k into the one for S_k+1:
  + 4 ? ^{(k+1) - 1}= 
 Collect the terms on the left-hand side over a common denominator:
  = 
 Expand the numerator of the left-hand side and simplify to get:
  = 
Since the equality holds, the statement is true for n = k + 1 as long as it is true for n = k. Furthermore, the statement is true for n = 1. Thus, the statement is true for all natural numbers n.