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Sequences, Series, and the Golden Ratio

Sequences, Series, and the Golden Ratio. by Bobby Stecher mark.stecher@maconstate.edu. Arithmetic Sequences. 1, 2, 3, 4, … 5, 8, 11, 14, 17, … 2, -2, -4, -6, -8, …. Geometric Sequences. 1, 2, 4, 8, 16, … 2, 6, 18, 54, 162, … 20, 10, 5, 2.5, 1.25, .625, …. Miscellaneous Sequences.

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Sequences, Series, and the Golden Ratio

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  1. Sequences, Series, and the Golden Ratio by Bobby Stecher mark.stecher@maconstate.edu

  2. Arithmetic Sequences • 1, 2, 3, 4, … • 5, 8, 11, 14, 17, … • 2, -2, -4, -6, -8, …

  3. Geometric Sequences • 1, 2, 4, 8, 16, … • 2, 6, 18, 54, 162, … • 20, 10, 5, 2.5, 1.25, .625, …

  4. Miscellaneous Sequences Step 1 Step 2 Step 3 Step 4 The nth triangular number has n rows and n elements on the last row. When written as a sequence we have 1, 3, 6, 10, 15, … What are the next 3 in the sequence? What is the 12 term in the sequence?

  5. Triangular number as a sum =1 =1 + 2 =1 + 2 + 3 12th triangular number =1 + 2 + 3 + … +11 + 12

  6. The sum of an arithmetic sequenceArithmetic Series 1 + 2 + 3 + 4 + 5 + … + 96 + 97 + 98 + 99 + 100 =(1 + 100) + (2 + 99) + (3 + 98) + (4 + 97) + (5 + 96) + … +(50 + 51) =(101) + (101) + (101) + (101) + (101) + … +(101) =50 x (101) =5050

  7. What is the 12th triangular number? The 12th triangular number has the same value as the sum of the first 12 natural numbers. 1 + 2 + 3 + 4 + … + 10 + 11 + 12 12 = (1 + 12) Adding Consecutive triangular numbers. 2 + = = 1 + 3 + 5 + 7 4 = (1 + 7) =16 2

  8. Sequence of Squares n … n … … … … 1 4 9 16 n2 1 1 + 3 1 + 3 + 5 1 + 3 + 5 + 7 1 + 3 + 5 + … + (2n – 1)

  9. Fibonacci Sequence Fibonacci Sequence • 1,1,2,3,5,8,13,21,34,55,89,… • Each element is obtained recursively by adding the two previous elements. • The numbers in the Fibonacci sequence can be found in nature. Examples include rabbits, pineapples, bees, and flowers.

  10. Fibonacci Sequence 2 x 3 3 x 5 1 x 2 1x1 5 x 8

  11. Fibonacci Spiral A golden rectangle divided into squares to form a Fibonacci spiral.

  12. The Golden Ratio The ratio of the two consecutive elements in the Fibonacci sequence converges to a constant. This number is referred to as the golden ratio, φ or Φ, also known as phi.

  13. Fibonacci Sequence and Bunnies

  14. Geometric Series 1 + 2 + 4 + 8 + 16 + 32 + 64 = 128 – 1 8 16 1 4 2 64 32 1 – rn S = a1 1 – r

  15. GeometricSeries 2 + 6 + 18 + 54 + 162 1 – rn r = 3 S = a1 1 – r a1 = 2 1 – 35 S = 2 n = 5 1 – 3 1 – 243 S = 2 = 242 1 – 3

  16. Magic Squares Fill in the square using the numbers 1 through 9 exactly once so that each row, column, and both diagonals add to the same amount. http://illuminations.nctm.org/lessons/6-8/magic/MagicSquares-AS-Uncovering.pdf

  17. Magic Squares Fill in the square using the numbers 1 through 16 exactly once so that each row, column, and both diagonals add to the same amount.

  18. Magic Squares Fill in the square using the numbers 1 through 25 exactly once so that each row, column, and both diagonals add to the same amount.

  19. Multiplying Magic Squares The Math Forum at Drexel http://mathforum.org/alejandre/magic.square/adler/adler.AxB.html

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