Use mathematical induction to prove that the statement is true for every positive integer n.(53)n = 53n

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 (53)(1) = 1251 = 125. 
 The right-hand side becomes 53(1) = 53 = 125. Thus, the statement is true for n = 1.
b). Assume the statement is true for n = k:
 (53)k= 53k
 Multiply both sides by 53:
  53(53)k = 53k ? 53
 Using the product rule for exponents and the distributive property,
 (53)k + 1 = 53k + 3 = 53(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.