Use mathematical induction to prove that the statement is true for every positive integer n.2n > 2n + 2, for n ? 4
What will be an ideal response?
Answers will vary. One possible proof follows.
a). Let n = 4. Then 24 > 2?4 + 2. So, the statement is true for n = 4.
b). Assume S is true for n = k:
2k > 2k + 2
Multiply both sides by 2 to obtain:
2?2k > 2(2k + 2), or
2k+1 > 4k + 4.
Re-write the right-hand side as 2(k + 1) + 2k + 2:
2k+1 > 2(k + 1) + 2k + 2.
Since k > 0, then 2(k + 1) + 2k + 2 > 2(k + 1) + 2. So,
2k+1 > 2(k + 1) + 2.
The statement is true for n = k + 1 if it is true for n = k. Furthermore, the statement is true for n = 4. Therefore, the statement is true for all n ? 4.
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