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Math 1536 Chapter 3: Proportion and the Golden Ratio The Game Of Fim The Game Start with any number of coins on the table. Two players take turns removing coins. Whoever removes the last coin wins. Rules

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math 1536

Math 1536

Chapter 3:Proportion and the Golden Ratio

the game
The Game
  • Start with any number of coins on the table.
  • Two players take turns removing coins.
  • Whoever removes the last coin wins.
rules
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’.
sequences
Sequences
  • A sequence is simply a list of numbers.
  • A sequence can be finite or infinite.
  • Our focus will be on infinite sequences.
infinite sequences
Infinite Sequences
  • We’ll cover three types of infinite sequences:
    • arithmetic
    • geometric
    • Fibonacci
arithmetic sequence
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.
example 1
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.
example 2
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.
skipping ahead
“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.
example 3
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.
example 3 continued 500th term of 7 10 13 16 19
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.
arithmetic sequence formula
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

geometric sequence
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.
example 4
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!
example 5
Example 5
  • However, geometric sequences can also get smaller if the number we’re multiplying is a fraction:

{8, 2, 0.5, 0.125, 0.03125, …}

example 6
Example 6
  • It’s also possible to ‘skip ahead’ in geometric sequences.
  • Find the 80th term of:

{2, 6, 18, 54, 162, 486, …}

example 6 continued 80th term of 2 6 18 54 162 486
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.
geometric sequence formula
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)

geometric series
Geometric Series
  • A geometric series is the sum of the terms in a geometric sequence:Sequence: {16, 8, 4, 2, …} Series: 16 + 8 + 4 + 2 + ···
sum of a geometric series
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)
graphs of sequences
Graphs of sequences
  • Before we begin studying the Fibonacci sequence, consider the graphs of the sequences we’ve just covered…
correspondence
Correspondence
  • There’s a correspondence between the algebraic aspects of the sequence and the visual aspects of the graph:
geometric sequence26
Geometric Sequence
  • The graph of a geometric sequence is an exponential graph.
fibonacci sequence
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.
fibonacci sequence cont
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,…}

fibonacci decomposition
Fibonacci Decomposition
  • Every positive integer has a decomposition into nonconsecutive Fibonacci numbers.
example 7
Example 7
  • The Fibonacci decomposition of 30 is: 30 = 21 + 8 + 121, 8, and 1 are all Fibonacci numbers, but no two of them appear side-by-side in the sequence.
ratios of the fibonacci sequence
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:
ratios of the fibonacci sequence33
Ratios of the Fibonacci Sequence
  • 1/1 = 1
  • 2/1 = 2
  • 3/2 = 1.5
  • 5/3 = 1.6667
  • 8/5 = 1.6
  • 13/8 = 1.625
  • 21/13 = 1.6154
  • 34/21 = 1.6191
ratios of the fibonacci sequence34
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.
the golden ratio
The Golden Ratio
  • The value at which they stabilize is (1 + 5)/2, which is approximately 1.618.
  • This value is known as the golden ratio, and is often abbreviated with the Greek letter  (phi).
the golden ratio37
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…