Using Differences to Identify Patterns

Section 1.1

• A _______ _________ is a string of numbers, or terms, in a certain order.

• If the difference from one term to the next in a number sequence is always the ______ the difference is called a _________ ___________.

number sequence

sameconstant difference

Example 1:

• Find the next three terms of each sequence by using constant differences.

A. 1, 3, 5, 7, 9, …

1 3 5 7 9 ___ ___ ___

+2 +2 +2 +2 +2 +2 +2

11 13 15

Example 1

B. 80, 73, 66, 59, 52, …

• 80 73 66 59 52 ___ ___ ___

-7 -7 -7 -7 -7 -7 -7

45 38 31

Try these…

C. 1, 4, 7, 10, 13, …

1 4 7 10 13 ___ ___ ___

+3 +3 +3 +3 +3 +3 +3

16 19 22

Try these…

D. 30, 25, 20, 15, 10, …

• 30 25 20 15 10 ___ ___ ___

-5 -5 -5 -5 -5 -5 -5

5 0 -5

Example 2

• Find the next three terms of each sequence by using constant differences.

E. 1, 4, 9,16, 25, …

1 4 9 16 25 ___ ___ ___

+3 +5 +7 +9 +11 +13 +15

36 49 64

+2 +2 +2 +2 +2+2

First differences

Second differences

Example 2

• F. 37, 41, 48, 58, 71, …

• 37 41 48 58 71

+4 +7 +10 +13 +16 +19 +22

87 106 128

+3 +3 +3 +3 +3+3

First differences

Second differences

Try these…

• G. Find the next three terms of each sequence by using constant differences.

• 2, 6, 12, 20, 30, …

• 2 6 12 20 30 ___ ___ ___

+4 +6 +8 +10 +12 +14 +16

+2 +2 +2 +2 +2+2

First differences

Second differences

42 56 72

Try these…

• H. Find the next three terms of each sequence by using constant differences.

• 8, 20, 30, 38, 44, …

• 8 20 30 38 44 ___ ___ ___

+12 +10 +8 +6 +4 +2 +0

-2 -2 -2 -2 -2-2

First differences

Second differences

48 50 50

• A___________ is a statement about observations that is believed to be true.

• Mathematicians try to prove or disprove conjectures.

• Let’s observe the next relationship and see if a conjecture can be made.

conjecture

Example 3• The table below shows the relationship between

temperatures in Celsius and temperatures in Fahrenheit. Use the method of constant differences to find the Fahrenheit temperatures that correspond to the Celsius temperatures of 50, 60, and 70.

Celsius 0 10 20 30 40 50 60 70

Fahrenheit 32 50 68 86 104

+18 +18+18 +18 +18 +18 +18

140 158122

• What conjecture can you make about this relationship?

For every 10 degrees that Celsius increases, the Fahrenheit increases 18 degrees.

The next three terms are _________________,

_________________, and ____________________.

• Some sequences can also be studied with diagrams.

For example, the sequence 2, 6, 12, 20, 30, … is found by counting the number of dots in the pattern below.

6 7

1 22 2 3

6 3 412 4 5

20 5 630

7 8 8 9

42

56 72

Problem solving strategies can include:

• Drawing a diagram

• Solving a simpler problem

• Making a table or chart

• Looking for a pattern

Example 4

• Suppose that 10 friends have just returned to school. Each friend has exactly one conversation with each of the other friends to talk about what they did during summer break. Use problem-solving strategies to determine how many conversations there will be.

one two three fourperson people people people

Arrange the information from the simpler problems in a table. Look for a pattern.

People 1 2 3 4 5 6 7 8 9 10

Conversations 0 1 3 6 10 15 21 28 36 45

65432 71 8 9

Use differences to determine how the number of conversations is increasing. Then extend the pattern to 10 people.

With 10 friends, it takes _______ conversations for each person to have exactly one conversation with each other person.

45