sequence an ordered list of numbers finding patterns
DESCRIPTION
Sequence an ordered list of numbers Finding Patterns. Arithmetic Geometric. Arithmetic Sequence. Pattern adding a fixed number from one term to the next COMMON DIFFERENCE d. a n =. NOTE n increases by only 1 at a time. n. Examples - PowerPoint PPT PresentationTRANSCRIPT
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Sequencean ordered list of numbers
Finding Patterns ArithmeticGeometric
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Arithmetic Sequence
Pattern adding a fixed number from one term to the next
COMMON DIFFERENCEd
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NOTEn increases byonly 1 at a time n
an =
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Examples
Sequence A: 5 , 8 , 11 , 14 , 17 , ... Sequence B: 26 , 31 , 36 , 41 , 46 , ... Sequence C: 20 , 18 , 16 , 14 , 12 , ...
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Common Difference - the fixed numbers that binds the sequence together
In Sequence A the common difference is +3
In Sequence B the common difference is +5
In Sequence C the common difference is -2
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Common Difference = d Generic sequence is referred to using the letter a along with subscripts as follows:
Generic Sequence: a1, a2, a3, a4, ...
The fifth term of a given sequence = a5.
The 17th term = a17
The nth term = an The term right before the nth term = an-1
d can be calculated by subtracting any two consecutive terms in an arithmetic sequence.
d = an - an - 1, where n is any positive integer greater than 1.
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Common difference (d) = 3
an = dn + c or an = 3n + cc is some number that must be found
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Common Difference = 5Formula for the nth term = an = 5n + 21
14th terma14 = 5(14) + 21 = 70 + 21 = 91
40th terma40 = 5(40) + 21 = 200 + 21 = 221
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Common difference = -2. Formula will be -2n + cFind c
-2×1 + c = -2 + c-2×2 + c = -4 + c-2×3 + c = -6 + c-2×4 + c = -8 + c-2×5 + c = -10 + c
C = 22
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Geometric Sequence
Pattern Multiply a fixed number from one term to the next
Common Ratio r
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an=
n
NOTEn increases byonly 1 at a time
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a1 a2 a3 a4
5, 10, 20, 40, ...multiply each term by 2 to arrive at the next term
or...divide a2 by a1 to find the common ratio, 2
r = 2
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-11, 22, -44, 88, ...
multiply each term by -2 to arrive at the next term
or...divide a2 by a1 to find the common ratio, -2
r = -2
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Find the common ratio for the sequence
r = divide the second term by the first term = -1/2.
Checking shows that multiplying each entry by -1/2 yields the next entry.
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Find the 7th term of the sequence2, 6, 18, 54, ...
n = 7; a1 = 2, r = 3
The seventh term is 1458.
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Find the 11th term of the sequence 1 1 1
1, , ,2 4 8
n = 11; a1 = 1, r = -1/2
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A ball is dropped from a height of 8’. The ball bounces to 80% of its previous height with each bounce.
How high (to the nearest tenth of a foot) does the ball bounce on the fifth bounce?
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Set up a model drawing for each "bounce". 6.4, 5.12, ___, ___, ___ The common ratio is 0.8.
Answer: 2.6 feet
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Finding More Patterns Fractal
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Carl Friedrich GaussGermany(1777 - 1855)
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Add the integers from 1 to 100
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There are 50 pairs of 101...
Add the integers from 1 to 100
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Do you think we can find a formula that will work for adding all the integers from 1 to n?
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How many pairs of n + 1 are there? Half of n!
5050 + 101 = 5151
Do you think we can find a formula that will work for adding all the integers from 1 to n?
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Write this series in sigma notation?
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A similar formula works for when the terms skip some numbers, like
To :
Find the sum of the first n terms
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This is for arithmetic sequences ONLY!Let's find the sum of the first 50 terms of the arithmetic sequence
We need:
We have:
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Finding More Patterns Fractal
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Koch’s SnowflakeFractal Pattern
What is its perimeter?
Start with an equilateral triangle.To each side….
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Koch’s SnowflakeFractal Pattern
Divide each side into thirds…The side length of each successive small triangle is 1/3
What is its perimeter?
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Koch’s SnowflakeFractal
Total length increases by one third and thus the length at step n will be (4/3)n
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