2.7 Cauchy Sequences 12110. (Project) Recursive Weighted Arithmetic Means: Let a -=j; b be
arbitrary real numbers, let 0 < t < 1, and define the sequence { xn} by
Xi= a , X2 = b, and '<In EN, Xn+2 = txn + (1 - t)xn+i·
That is, each new term beginning with the third is a weighted average
of the two previous terms. Geometrically, Xn+ 2 is a point in the interval
between Xn and Xn+i that cuts the interval into two segments whose
lengths are in the ratio t to 1 -t. Prove that { Xn} is contractive (defined
in Exercise 8), and find its limit.- Contraction Mappings: Let a< band I= [a, b]. A function f: I~ I
is said to be a contraction mapping if 3 c 3 0 < c < 1 and '<Ix, y E I,
lf(x) - f(y)I ::; clx - YI· Prove that a contraction mapping must have at
least one "fixed point," x E I 3 f(x) = x. [See Exercise 8.] Also prove
that f cannot have more than one fixed point in J. - (Project) Fibonacci Numbers: The Fibonacci sequence consists of
the Fibonacci numbers, 1, 1, 2, 3, 5, 8, 13 , 21 , ···,and is defined recursively
by Ji = 1, h = 1, and '<In ::'.'.: 2, fn+2 = fn+i + fn· Each new term
after the second is the sum of the previous two terms. Many interesting
results have been proved about the Fibonacci numbers- enough to fill an
entire book. We shall be concerned here with the sequence of ratios of
successive Fibonacci numbers. We begin by defining the sequence {rn}
b
fn+i
Y rn = fn.
(a) Develop a table that shows the first 10 terms of {rn}· On the basis of
this table, conjecture answers to the questions: Does {rn} converge?
Is it monotone? Eventually monotone? Can you find a strictly
increasing subsequence? A strictly decreasing subsequence? (No
proofs required.)
1
(b) Prove that '<In EN, rn+l = 1 + -.
rn
(c) Prove that '<In::'.'.: 2, ~ < rn < 2.
(d) Prove that {rn} is "contractive,'' and hence is a Cauchy sequence.
(e) Find lim rn. [Keep note of this limit; it will reappear.]
n-+oo... l+v's
(f) The quadratic equation x^2 -x-1 = 0 has two solutions, a = --
2
1 - v's
and (3 = -
2
-. Show that a + (3 = 1, a^2 = a + 1, and (3^2 = (3 + 1,
and from these facts show that '<In E N, an+^2 = an+I +an and
(3n+2 = (3n+l + (3n.