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39 votes
39 votes
#2
#3
Need help for my discrete math class

#2 #3 Need help for my discrete math class-example-1
User Augustus Kling
by
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1 Answer

23 votes
23 votes

For (2), start with the base case. When n = 2, we have

(n + 1)! = (2 + 1)! = 3! = 6

2ⁿ = 2² = 4

6 > 4, so the case of n = 2 is true.

Now assume the inequality holds for n = k, so that

(k + 1)! > 2ᵏ

Under this hypothesis, we want to show the inequality holds for n = k + 1. By definition of factorial, we have

((k + 1) + 1)! = (k + 2)! = (k + 2) (k + 1)!

Then by our hypothesis,

(k + 2) (k + 1)! > (k + 2) 2ᵏ = k•2ᵏ + 2ᵏ⁺¹

and k•2ᵏ ≥ 2•2² = 8, so

k•2ᵏ + 2ᵏ⁺¹ ≥ 8 + 2ᵏ⁺¹ > 2ᵏ⁺¹

which proves the claim.

Unfortunately, I can't help you with (3). Sorry!

User Fahadkalis
by
2.5k points
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