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Chapter 9
Notes from________
Important vocabulary terms:
Sequence: An ordered list of numbers that is generated by a rule. Generally sequences are notated
with a capital S.
Terms: The numbers in the list are called the terms of the sequence.
Term Number: (usually denoted “N”) is the number that tells the term’s relative position in the
sequence list.
Recursive Rule: the rule that tells how to get the next term of the sequence using earlier terms in
the sequence.
Explicit Rule: the formula that generates the sequence from the term number.
Recursive rules are usually easier to understand and more convenient when you are writing out the
terms of the sequence in order. BUT recursive rules don’t help if you want to know the 1000th term
without finding the other 999 other terms first.
Example 1: The odd counting numbers form a sequence:
S N = {1, 3, 5, 7, 9, . . . }
The recursive rule for the odd counting numbers is:
a) In words ______________________________________________________
b) As an equation ___________________________
c) The explicit rule for the odd counting numbers is SN = _______________________________
= _______________________________
2. Use the sequence rule SN = 2N – 1 to first describe in words and then find each value
a) S3 ____________________________________________________ ______________________
b) 4S5 ___________________________________________________ ________________________
c) S3 + 4 ___________________________________________________ ________________________
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Arithmetic Sequences have a common difference between terms
Implicit Rule:
Start with a1
and add d each time
Explicit Formula:
an= a1 + (n – 1)d
5, 10, 15, 20, _____, _____, ______, . . .
18, 8, -2, -12, _____, _____, ______, . . .
Geometric Sequences have a common ratio between terms:
Implicit Rule:
Start with a1 and
multiply by r each time
Explicit Formula:
an = a1r(n -1)
2, 4, 8, 16, _____, ______, ______, . . .
80, 40, 20, 10, _____, _____, ______, . . .
3. a) Write the first five terms of the sequence that has the recursive rule “start with 2. For each successive
term, multiply the previous term by 3”.
________, _________, _________, ___________, ___________
b) Is the sequence above arithmetic or geometric? _________________
4. a) Write out the first 5 terms of the sequence Sn = 5n – 1 in list form.
(Hint, remember you start with n = 1)
______, _______, _________, __________, _______________
b) What is the numerical value of the 500th term in this sequence? _______
c) Is the sequence above arithmetic or geometric? ___________________
5. As you are watching the video clip, write down the Fibonacci Sequence.
_______________________________________________________________________________
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Fibonacci Numbers are not only related to patterns in nature, they form fascinating mathematical
patterns and properties as well. These will be the subjects of the rest of our study of Fibonacci Numbers.
Pattern #1: Every counting number (Fibonacci or not) can be written as the sum of distinct Fibonacci
numbers. “Distinct” here means that you cannot use the same number more than once. Each of the numbers
to be added must be a different number from all the rest.
In fact, there is usually more than one way to do it.
6. a) Given F6 = 8. Write 8 as the sum of distinct Fibonacci numbers in two different ways.
7. Consider the following sequence involving Fibonacci numbers:
2(2) – 3 = 1
2(3) – 5 = 1
2(5) – 8 = 2
2(8) – 13 = 3
Look for the pattern and then write down a reasonable choice for the fifth equation in this sequence as a
recursive equation and in words.
__________________________________
________________________________________________________________________________________
________________________________________________________________________________________
8. Compute each of the following:
a) F14 + 1 __________________________________________
b) F14 + 1 ___________________________________________
c) F14/2______________________________________________
d) 6F4 – 2 ____________________________________________
Assignment Due Fri. 9/26 Read Chapter 9 pp. 312-316, Finish Guided Notes
Do #1-4, 6, 7, 8, 13, on pp. 329-330
& make Test Corrections (on a separate sheet of paper) Due Oct. 1
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