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Chapter 9 - Number patterns and recursion - Coggle Diagram
Chapter 9 - Number patterns and recursion
8A - Terminology
Sequences
A list of numbers
Term --> each number in the sequence
Sequences can be randomly generated or generated by a rule
The dots (...) at the end of a sequence indicate it is infinite
Arithmetic (linear) sequence -- Add or subtract a constant number to get the next term
Geometric sequence -- multiple or divide a constant number to get the next term
The meaning of tn
Used to denote the nth term and therefore the rule of the sequence.
t1, t2, t3, are used as labels for the first three terms in the sequence
8B - The common difference
In an arithmetic sequence the fixed number (added or subtracted) from each term to make the next term is called the common difference (d)
d = any term - its previous term
8C - The rule for arithmetic sequences
t1 = a+(n - 1)d
a = t1 = first term
d = common difference
n = position number in the term
8D - Another way of looking at the rule (arithmetic sequences)
Recurrence relation --> a rule that we can use to generate a new sequence (always has a starting point and a rule)
8E - The common ratio
The common ratio is found by dividing the next term by the current term. (t2 / t1)
r > 0
Graphs of geometric systems for r > 0
Increasing when r is greater than 1
decreasing when r is less than one and when r = 0
The value of the common ration (multiplies terms)
The graph of the term number vs term value is curved
8F - Finding the nth term
The rule for finding the nth term of a geometric sequence is tn = ar^n-1
Percentage change = Worded questions often use percentages therefore we need to know how to convert a percentage into a common ratio
For a reduction -- r = 1 - % / 100
For an increase -- r = 1 + %/100
8G - Another way (geometric sequences)
The general recurrence relation is -- t1 = a
The recurrence rule can be used to generate an arithmetic sequences -- tn+1 = tn x r
The general rule for the nth term is the best in finding a specific term ( tn = ar^n+1)
8H - Linear growth and decay
Linear growth or decay in a sequence occurs when a quantity increases or decreases by the same amount at regular intervals
Everyday examples include: Simple interest or the deprecation of the cost of a car
Linear growth / decay rule
For growth
V0 -- starting or initial value (Vn+1 = Vn _ D
Vn = V0 + n x D
For decay
Vn = V0 - n x D
8I - The Fibonacci sequence
After the first two terms each term is the sum of the preceding two terms
The recurrence relation that generates the Fibonacci sequence is an example of second order recurrence relation. It links terms that are two steps apart.