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### Chapter 3

##### Section 3.1

1. a. 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025.

c. 64, 32, 16, 8, 4, 2, 1, 1/2, 1/4, 1/8.

e. 1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89.

2. a. Basis: 1 ∈ S; Induction: If xS, then x + 2 ∈ S.

c. Basis: –3 ∈ S; Induction: If xS, then x + 2 ∈ S.

e. Basis: 1 ∈ S; Induction: If xS, then ($\sqrt{x}$ + 1)2S.

3. a. Basis: 4, 3 ∈ S. Induction: If xS, then x + 3 ∈ S.

4. a. Basis: 0, 1 ∈ S; Induction: If xS, then x + 4 ∈ S.

c. Basis: 2 ∈ S; Induction: If xS, then x + 5 ∈ S.

5. 4 = 3 ∪ {3} = 2 ∪ {2} ∪ {3} = 1 ∪ {1} ∪ {2} ∪ {3} = 0 ∪ {0} ∪ {1} ∪ {2} ∪ {3} = Ø ∪ {0} ∪ {1} ∪ {2} ∪ {3} = {0, 1, 2, 3}.

6. a. Basis: bS. Induction: If xS, then axcS.

c. Basis: aS. Induction: If xS, then aaxS.

e. Basis: ...

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