Use mathematical induction to prove that the statement is true for every positive integer n.(72)n = 72n

Question

What will be an ideal response?

Answer 1

Answers will vary. One possible proof follows.
a). Let n = 1. The left-hand side becomes (72)(1) = 491 = 49. 
 The right-hand side becomes 72(1) = 72 = 49. Thus, the statement is true for n = 1.
b). Assume the statement is true for n = k:
 (72)k= 72k
 Multiply both sides by 72:
  72(72)k = 72k ? 72
 Using the product rule for exponents and the distributive property,
 (72)k + 1 = 72k + 2 = 72(k + 1)
The statement is true for n = k + 1 as long as it is true for n = k. furthermore, it is true for n = 1. Thus, the statement is true for all natural numbers n.