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Section 1.2
Patterns and Problem Solving
Arithmetic Sequence
An arithmetic sequence has a common
difference between each of the numbers in
the sequence.
Examples:
1, 3, 5, 7, 9, …
1, 2, 3, 4, 5, …
2, 4, 6, 8, 10, …
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Geometric Sequence
A geometric sequence is formed by
multiplying the previous number by a
selected number. Therefore, there is a
common ratio from one number to the next
in a geometric sequence.
Examples:
2, 4, 8, 16, 32, 64, …
3, 9, 27, 81, 243, …
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22)Identify each of the
sequences as arithmetic or
geometric. State a rule for
obtaining each number from
the preceding number.
What is the 12th number in
each sequence?
a) 280, 257, 234, 211,...
b) 17, 51, 153,459,…
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Method of Finite Differences
Finding number patterns by looking at the
differences between consecutive terms.
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26)Use the method of finite
differences to find the next
number in each sequence.
a) 3, 7, 13, 21, 31, 43,…
b) 215, 124, 63, 26, 7, …
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Fibonacci Numbers
After the first two numbers, 1 and
1, each successive number is
obtained by adding the previous
two.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
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14) A Fibonacci-type sequence can be
started with any two numbers.
Find the missing numbers among
the first 10 numbers of the
Fibonacci-type sequences.
a) 10, ___, 24, ___, ___, 100, ___, ___, ___, 686
b) 2, ___, ___, 16, 25, ___, ___, ___, ___, 280
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14) continued
c) The sum of the first 10 numbers in the
sequence in part a) is equal to 11 times
the seventh number, 162. What is this
sum?
d) Can the sum of the first 10 numbers in
the sequence in part b) be obtained by
multiplying the seventh number by 11?
e) Do you think the sum of the first 10
numbers in any Fibonacci-type sequence
will always be 11 times the seventh
number? Try some other Fibonacci-type
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sequences to support your conclusion.
The previous problem is an example of
inductive reasoning.
Inductive reasoning is the process of
forming conclusions on the basis of
patterns, observations, examples,
experiments. Otherwise known as an
informed guess.
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Pascal’s Triangle
Row 0
1
2
3
4
5
6
.
.
1
1
1
1
1
2
3
1
3
1
1 4
6
4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
.
.
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20)Compute the sums of the
numbers in the first few rows
of Pascal’s triangle. What
kind of sequence (arithmetic
or geometric) do these sums
form?
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A counterexample is an example
that shows a statement to be false.
40) Find a counterexample for each of the
statements.
a) Every whole number greater than 4 and less
than 20 is the sum of two or more
consecutive whole numbers.
b) Every whole number between 25 and 50 is
the product of two whole numbers greater
than 1.
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