1. Chapter 3: Proportion and the Golden Ratio
Math 1536
Chapter 3: Proportion and the Golden Ratio

2. The Game Of Fim

3. The Game Start with any number of coins on the table.
Two players take turns removing coins.
Whoever removes the last coin wins.

4. Rules
Whoever plays first may take any number of coins (but not all of them on the first turn; that wouldn’t be fair!)
From then on, each player may take up to twice the last number played. (Example: If your opponent just took three coins, you may take up to six.)
You must take at least one coin on your turn—no ‘passing’.

5. Sequences A sequence is simply a list of numbers.
A sequence can be finite or infinite.
Our focus will be on infinite sequences.

6. Infinite Sequences We’ll cover three types of infinite sequences:
arithmetic
geometric
Fibonacci

7. Arithmetic sequence
An arithmetic sequence is a sequence in which each term is found by adding a constant value to the previous term. Every term grows by the same amount.

8. Example 1 The following sequence is arithmetic:
{5,9,13,17,21,25,29,…}
The starting point is 5, and each term increases by 4.

9. Example 2 An arithmetic sequence may also decrease:
{60,53,46,39,32,25,18,…}
In this example, the starting point is 60, and the terms all decrease by 7.

10. “Skipping Ahead”
Sometimes, we’ll want to compute a distant term of the sequence directly without having to compute all the intervening terms.
For example, we’d like to be able to find the 500th term without having to write them all out.

11. Example 3 Find the 500th term of this arithmetic sequence:
{7,10,13,16,19,22,…}
This sequence starts at 7 and increases by 3.

12. Example 3 (continued) 500th term of {7,10,13,16,19…}
To skip from the first term to the 500th term, we have to skip over 499 terms. Each term added 3, for a total of 499  3 =1497.
Add this to the starting value (7) to get 1504.

13. Arithmetic Sequence Formula
To find the nth term of an arithmetic sequence that starts with the value a and increases by r each term:
a + (n – 1)r

14. Geometric Sequence
A geometric sequence differs from an arithmetic sequence in that each term is obtained by multiplying the previous term by a constant value.

15. Example 4 The following sequence is geometric:
{7,35,165,825,4125,20625,…}
The starting point is 7, and each term is found by multiplying by 5.
Geometric sequences can get very big very fast!

16. Example 5
However, geometric sequences can also get smaller if the number we’re multiplying is a fraction:
{8, 2, 0.5, 0.125, , …}

17. Example 6 It’s also possible to ‘skip ahead’ in geometric sequences.
Find the 80th term of:
{2, 6, 18, 54, 162, 486, …}

18. Example 6 (continued) 80th term of {2, 6, 18, 54, 162, 486, …}
To get from the first term to the 80th, we have to skip over 79 terms. Each term increases by multiplying by 3, for a total of 3333… (79 times) = 379.
Multiply by the starting point to get 2379.

19. Geometric Sequence Formula
To find the nth term of a geometric sequence that starts at a and grows by multiplying by r each term:
a  r(n – 1)

20. Geometric Series
A geometric series is the sum of the terms in a geometric sequence: Sequence: {16, 8, 4, 2, …} Series: ···

21. Sum of a Geometric Series
A geometric series is meaningful only if |r| < 1 (so that the terms are getting smaller). In this case, the total is equal to: a / (1–r)

22. Graphs of sequences
Before we begin studying the Fibonacci sequence, consider the graphs of the sequences we’ve just covered…

23. Arithmetic Sequence {2,5,8,11,14,17,…}

24. Correspondence
There’s a correspondence between the algebraic aspects of the sequence and the visual aspects of the graph:
Algebraic
Visual
starting point (a)
y-intercept (height)
growth (r)
slope

25. Geometric Sequence {1,2,4,8,16,32,…}

26. Geometric Sequence
The graph of a geometric sequence is an exponential graph.

27. Fibonacci Sequence
The Fibonacci sequence is a hybrid of sorts -- it grows by adding, like an arithmetic sequence, but does so in a way that it increases very fast, like a geometric sequence.

28. Fibonacci Sequence (cont.)
The Fibonacci sequence begins with 1, 1, and then each succeeding term is found by adding up the two previous terms:
{1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…}
Some sources show the Fibonacci sequence beginning with 0, 1 instead of 1,1. This has no effect on the rest of the sequence.

29. Graph of the Fibonacci Sequence

30. Fibonacci Decomposition
Every positive integer has a decomposition into nonconsecutive Fibonacci numbers.

31. Example 7
The Fibonacci decomposition of 30 is: = , 8, and 1 are all Fibonacci numbers, but no two of them appear side-by-side in the sequence.

32. Ratios of the Fibonacci Sequence
If we form fractions of terms of the Fibonacci sequence -- taking each pair of numbers in turn and placing the higher number over the lower number, an interesting pattern emerges:

33. Ratios of the Fibonacci Sequence
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 =
8/5 = 1.6
13/8 = 1.625
21/13 =
34/21 =

34. Ratios of the Fibonacci Sequence
The ratios alternately increase, then decrease, but they quickly stabilize around a single value.
See the graph on the next page.

35. Graph of the Ratios

36. The Golden Ratio
The value at which they stabilize is (1 + 5)/2, which is approximately
This value is known as the golden ratio, and is often abbreviated with the Greek letter  (phi).

37. The Golden Ratio
The Golden Ratio is the only positive number that makes the equation 1 + x = x2 true.
(Use the quadratic formula to show this!)
We’ll use this fact next time…