fun with sequences and series tim jehl – math dude
TRANSCRIPT
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FUN WITHSEQUENCES AND SERIES
TIM JEHL – MATH DUDE
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THE PYRAMIDNumber of levels
Blocks in last level
Blocks in pyramid
1 1 1
2 2 3
3 3 6
4 4 10
5 5 15
6 6 21
7 7 28
8 8 36
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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
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THE PYRAMID IN 3DNumber of levels
Blocks in last level
Blocks in pyramid
1 1 1
2 4 5
3 9 14
4 16 30
5 25 55
6 36 91
7 49 140
8 64 204
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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
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NESTED SQUARES
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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)
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FIBONACCI
0
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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
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SIERPENSKI’S TRIANGLE
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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
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SIERPENSKI’S TRIANGLE REVISITED
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VON KOCH SNOWFLAKE
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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
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PLUS
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GOLDEN RATIO