MATH 2160 Sequences. Arithmetic Sequences The difference between any two consecutive terms is always...
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Transcript of MATH 2160 Sequences. Arithmetic Sequences The difference between any two consecutive terms is always...
MATH 2160
Sequences
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, …
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
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
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
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, …
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
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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1
11
The Ultimate Pattern…
Fibonacci Sequence
Rabbit Breeding Pattern(# of Pairs)
The Golden Rectangle
The Golden Ratio
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