5.5 the fibonacci sequence and the golden ratio the fibonacci sequence is the sequence … · 2015....
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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
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