1549901369-Elements_of_Real_Analysis__Denlinger_

122 Chapter 2 • Sequences

(g)

(h)

an - (Jn

Vn EN, define Un = , where a and (3 are as defined in (f).

a -(3

Prove that u 1 = 1, U2 = 1, and Vn 2 2, Un+2 = Un+I +Un. Thus,

{ un} must b e the Fibonacci sequence. We have found an explicit

formula for the Fibonacci numbers: fn =Un.

Geometric significance of a. Consider a rectangle whose width a

and length a + b are so proportioned that when a square of side a

is removed, as shown here, the remaining rectangle has width and

... a+b a
length m the same proportion. That 1s, - a- = b.

a b

a a a

a b

Figure 2.7

The classical Greek mathematicians called this ratio R = ~

b

the "Golden Ratio," and any rectangle with sides in this ratio a

"Golden rectangle." They considered it to be the most aestheti-

cally pleasing of a ll rectangles, and used it frequently in their art and

architecture. Prove algebraically that R = a, defined in (f) above.

(i) Prove that Vn 2 2, fn+ifn-1 - (fn)^2 = (-l)n.

(-l)n+l

(j) Prove that Vn E N, rn+I - rn = f f

n n+l

(k) Use (j) to prove that {r2n} is strictly decreasing and {r 2 n+i} is

strictly increasing.

1

13. Let a 2 1. Define the sequence {xn} by x 1 = a , and Xn+I = a+ -.

Xn

1.

Prove that Vn 2 2, a +

2

a :::; Xn :::; 2a, and use this result to prove that

{ Xn} is contractive. Find lim Xn.

n->oo

1

14. Let a > 1. Define the sequence {xn} by X1 = a, and Xn+I =

a+xn

1

Prove that Vn EN,

2

a :::; Xn :::; a , and use this result to prove that {xn}

is contractive. Find lim Xn· Compare this limit with that of Exercise 13.

n->oo