Use mathematical induction to prove that the statement is true for every positive integer n.(22)n = 22n

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 (22)(1) = 811 = 81 . The right-hand side becomes 34(1) = 34 = 81. Thus, the statement is true for n = 1.
b). Assume the statement is true for n = k:
 (34)k= 34k
 Multiply both sides by 34:
  34(34)k = 34k?34
 Using the product rule for exponents and the distributive property,
 (34)(k+1) = 34k+4 = 34(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.