5.5 The Fibonacci Sequence and the Golden Ratio

The Fibonacci Sequence is the sequence

1, 1, 2, 3, 5, 8, 13, ... .

After the first two terms (1 and 1), each term is obtained

by adding the two previous terms.

Recursive Formula for the Fibonacci Sequence

If Fn represents the Fibonacci number in the nth position

in the sequence, then

1

2

2 1

1

1

, for 3.n n n

F

F

F F F n

Example. Given that F18 = 2584 and F19 = 4181, find

(a) F17

(b) F20

The Golden Ratio

Consider the quotients of successive Fibonacci numbers

and notice a pattern.

These quotients approach 1+√5

2 ≈ 1.618, which is known

as the golden ratio.

1 2 3 51, 2, 1.5, 1.66...,

1 1 2 3

8 13 211.6, 1.625, 1.615384

5 8 13

A golden rectangle is one whose length and width are in

the golden ratio.

The golden rectangle appears frequently in art and

architecture; have a look at pages 211-212 of your text

for some examples.

Also:

Pineapple

Chambered Nautilus

Storm

Parthenon Mona Lisa

Example. Find the length of the long side of a golden

rectangle whose shorter side has a length of 34 inches.

34 in

L

7.5 Exponents and Scientific Notation

Exponential Expressions

Properties: For 𝑎 ∈ ℝ 𝑎𝑛𝑑 𝑚, 𝑛 ∈ ℕ:

𝑎𝑚𝑎𝑛 =

𝑎𝑚

𝑎𝑛=

(𝑎𝑚)𝑛 =

(𝑎𝑏)𝑚 =

(𝑎

𝑏)

𝑚

=

More Properties: For 𝑎 ∈ ℝ 𝑎𝑛𝑑 𝑚, 𝑛 ∈ ℕ:

𝑎0 =

0𝑛 =

𝑎1 =

1𝑛 =

𝑎−1 =

00 =

1

𝑎−1=

Example. Evaluate each expression.

(a) (–2)4

(b) –24

Example. Evaluate each expression.

(c) 4−2

5−3

(d) (𝑥2

𝑦3)−4

Scientific Notation

A number is written in scientific notation when it is

expressed in the form 𝑎 × 10𝑛,

with 1 ≤ | 𝑎 | ≤ 10 and n is an integer.

Example: 800 = 8 × 102

Example. Convert from standard notation to scientific

notation.

(a) 4,500,000

(b) 0.000034

Example. Convert from scientific notation to standard

notation.

(a) 1.97 × 105

(b) 3.08 × 10−3