ASSIGNMENT 2.1
There are two parts to this assignment. Part A is to be completed online before noon on Friday, January
14. Part B, which should be typeset (preferably with LaTeX), is to be submitted to the assignment box by the
same time.
Include your full name and student ID, along with the subject and assignment number, at the top of the front
page.
Part A [15 marks]
This part of the assignment can be found online, labelled Assignment 2.1, at www.mathxl.com. Questions
about the MathXL program itself should be sent to Eric; in fact, questions sent through the site will be directed
to him. To ask me questions, email me directly or come to my office.
Part B [15 marks]
The Fibonacci sequence {an } is defined by
an =
1
an−1 + an−2
for n = 1, 2
.
for n ≥ 3
Let {rn } be the sequence of ratios defined
by rn = an+1 /an . In this question, you will prove that {rn }
√
converges to ϕ, the golden ratio (1 + 5)/2.
(i) Verify the identities
rn = 1 +
1
for n ≥ 2
rn−1
and
rn = 2 −
1
for n ≥ 3.
rn−2 + 1
(ii) Prove that if {rn } converges, it converges to ϕ.
(iii) Show that if rn−2 ≥ ϕ, then rn ≥ ϕ; and if rn−2 ≤ ϕ, then rn ≤ ϕ.
(iv) Prove, by induction or otherwise, that the sequences {r2n−1 } and {r2n } are increasing and decreasing,
respectively.
(v) Prove that the assumption in part (b) is justified; that is, prove that {rn } converges.