Math 104: Homework 1 (due January 28)

1. Ross exercise 1.3

2. Ross exercise 1.8

3. The Leonardo numbers are defined according to L_{0} = 1, L_{1} = 1, and

L_{n+1} = L_{n} + L_{n−1} + 1

for all n ∈ . Consider the function

f(n) = \frac{2}{\sqrt{5}}(φ^{n+1} − (1 − φ)^{n+1} ) − 1

where φ = (1 + \sqrt{5})/2 is the Golden Ratio. For n ∈ , define P_{n} to be the proposition that “both L_{n} = f(n) and L_{n−1} = f(n − 1)”. Apply mathematical induction to prove that P_{n} is true for all n ∈ , and deduce that L_{n} = f(n) for all n ∈ ∪ {0}.

4. Ross exercise 2.4

5. Show that \sqrt{2}+ \sqrt{3} is irrational.

6. Ross exercise 3.1

7. Ross exercise 3.4

8. Show that ||a| − |b|| ≤ |a − b| for all a, b ∈ .

9. Ross exercise 3.6