math 2160 sequences. arithmetic sequences the difference between any two consecutive terms is always...
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MATH 2160
Sequences
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Arithmetic Sequences
The difference between any two consecutive terms is always the same. Examples:
1, 2, 3, … 1, 3, 5, 7, … 5, 10, 15, 20, …
Non-Examples 1, 4, 9, 16, … 2, 6, 12, 20, …
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Arithmetic Sequences
The nth number in a series: an = a1 + (n – 1) d
Example Given 2, 5, 8, …; find the 100th term
n = 100; a1 = 2; d = 3 an = 2 + (100 – 1) 3 an = 2 + (99) 3 an = 2 + 297 an = 299
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Arithmetic Sequences
Summing or adding up n terms in a sequence: Example:
Given 2, 5, 8, …; add the first 50 terms n = 50; a1 = 2; an = 2 + (50 – 1) 3 = 149 Sn = (50/2) (2 + 149) Sn = 25 (151) Sn = 3775
nn aan
S 12
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Arithmetic Sequences
Summing or adding up n terms in a sequence: Example:
Given 2, 5, 8, …; add the first 51 terms n = 51; a1 = 2; a2 = 5; an = 2+(51 –
1)3=152 Sn = 2+((51-1)/2) (5 + 152) Sn = 2+25 (157) Sn = 2+3925 Sn = 3927
n 1 2 n
n 1S a a a
2
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Geometric Sequences
The ratio between any two consecutive terms is always the same. Examples:
1, 2, 4, 8, … 1, 3, 9, 27, … 5, 20, 80, 320, …
Non-Examples 1, 4, 9, 16, … 2, 6, 12, 20, …
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Geometric Sequences
The nth number in a series: an = a1 r(n-1)
Example Given 5, 20, 80, 320, …; find the 10th term
n = 10; a1 = 5; r = 20/5 = 4 an = 5 (4(10-1)) an = 5 (49) an = 5 (262144) an = 1310720
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Geometric Sequences
Summing or adding up n terms in a sequence: Example:
Given 5, 20, 80, 320, …; add the first 7 terms n = 7; a1 = 5; r= 20/5 = 4 Sn = 5(1 – 47)/(1 – 4) Sn = 5(1 – 16384)/(– 3) = 5(– 16383)/(– 3) Sn = (– 81915)/(– 3) = (81915)/(3) Sn = 27305
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The Ultimate Pattern…
Fibonacci Sequence
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Rabbit Breeding Pattern(# of Pairs)
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The Golden Rectangle
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The Golden Ratio
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Fibonacci Sequences
1, 1, 2, 3, … Seen in nature
Pine cone Sunflower Snails Nautilus
Golden ratio (n + 1) term / n term of Fibonacci Golden ratio ≈ 1.618