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Fun with Sequences And Series

Fun with Sequences And Series. Tim Jehl – Math Dude. The Pyramid. Pyramid Math. Arithmetic sequence for the number of blocks in each level a n = n The total number of blocks in the pyramid is a series S n = (n)(n+1)/2. The Pyramid in 3D. 3D Pyramid Math.

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Fun with Sequences And Series

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  1. Fun withSequences And Series Tim Jehl – Math Dude

  2. The Pyramid

  3. Pyramid Math • Arithmetic sequence for the number of blocks in each level • an = n • The total number of blocks in the pyramid is a series • Sn = (n)(n+1)/2

  4. The Pyramid in 3D

  5. 3D Pyramid Math • Power sequence for the number of blocks in each level • an = n2 • The total number of blocks in the pyramid is a series • Sn = (n)(n+1)(2n+1)/6

  6. Nested Squares

  7. Nested Squares math • Complete the table below. The first square you drew corresponds to n = 1, the second square is n = 2, etc. • Side Length of nth square (Ln) Ln = L1(.707)n-1 • Side Length of nth square divided by 2 (Ln/2) (L1/2)(.707)n-1 • Perimeter of nth square (Pn) 4L1(.707)n-1 • nth partial sum of perimeters (Sn) 4L1 (1-.707n)/(1-.707) • Write a recursive formula for the perimeter of the nth square (Pn). Pn = .707Pn-1 • Write an explicit formula for the perimeter of the nth square (Pn). 4L1(.707)n-1 • Find the formula for the nth partial sum of the perimeters (Sn) 4L1 (1-.707n)/(1-.707) • If the series for the perimeters continues forever, what is the sum of the perimeters of all squares (S)?4L1/(1-.707)

  8. Fibonacci 0

  9. Fibonacci Math • Write the recursive formula for the Fibonacci Sequence; you will need to specify the first two terms (1 and 1). • Complete the following table, where fn is the nth term of the Fibonacci Sequence • fn • fn+1 • fn+1/fn • What value does fn+1 / fn approach as n gets bigger? This value is the golden ratio. • Take the golden ratio and subtract 1. Find the reciprocal of the golden ratio. Notice anything? • Take the golden ratio and add 1. Square the golden ratio. Notice anything? Pretty cool, huh? • Golden Ratio

  10. Sierpenski’s triangle

  11. Sierpenski’s triangle • Complete the following table. Assume that your original triangle had an area of 100 cm2 and that n =1 is the removal of the first triangle. • Number of triangles removed during iteration (tn) • Area of one of the removed triangles (An) • Area removed during iteration (tn x An) • Total area remaining in Seirpenski’s Triangle • Total number of triangles removed (i.e. upside down triangles) • Find a recursive formula for the area remaining in Seirpenski’s Triangle. • What is the area of Seirpenski’s Triangle after infinite iterations? • Find a recursive formula for the number of upside down triangles in Seirpenski’s Triangle after n iterations. • Number of triangles removed each time is geometric • an = 3n-1 • Area of the removed triangles is ¼ of the remaining area • Area remaining is ¾ of the area - an = (3/4)n-1

  12. Sierpenski’striangle revisited

  13. Von Koch snowflake

  14. Von Koch snowflake math • Complete the following table. Assume your first triangle had a perimeter of 9 inches. • Number of line segments (tn) • Length of each segment (Ln) • Perimeter of snowflake (Pn) • Write a recursive formula for the number of segments in the snowflake (tn). • Write a recursive formula for the length of the segments (Ln). • Write a recursive formula for the perimeter of the snowflake (Pn). • Write the explicit formulas for tn, Ln, and Pn. • What is the perimeter of the infinite von Koch Snowflake? • Can you show why the area of the von Koch Snowflake is sum 4n-3x3.5/32n-7

  15. PLUS

  16. Golden Ratio

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