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 same constant 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 ___ ___ ___ 11 13 15 +2 +2 +2 +2 +2 +2 +2

Example 1 B. 80, 73, 66, 59, 52, … 80 73 66 59 52 ___ ___ ___ 45 38 31 -7 -7 -7 -7 -7 -7 -7

Try these… 1, 4, 7, 10, 13, … 1 4 7 10 13 ___ ___ ___ 16 19 22 +3 +3 1 4 7 10 13 ___ ___ ___ 16 19 22 +3 +3 +3 +3 +3 +3 +3

Try these… D. 30, 25, 20, 15, 10, … 30 25 20 15 10 ___ ___ ___ 5 -5 -5 30 25 20 15 10 ___ ___ ___ 5 -5 -5 -5 -5 -5 -5 -5 -5

Example 2 Find the next three terms of each sequence by using constant differences. 1, 4, 9,16, 25, … 1 4 9 16 25 ___ ___ ___ 36 49 64 First differences +3 +5 +7 +9 +11 +13 +15 Second differences +2 +2 +2 +2 +2 +2

Example 2 F. 37, 41, 48, 58, 71, … 37 41 48 58 71 87 106 128 First differences +4 +7 +10 +13 +16 +19 +22 Second differences +3 +3 +3 +3 +3 +3

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 ___ ___ ___ 42 56 72 First differences +4 +6 +8 +10 +12 +14 +16 Second differences +2 +2 +2 +2 +2 +2

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 ___ ___ ___ 48 50 50 First differences +12 +10 +8 +6 +4 +2 +0 Second differences -2 -2 -2 -2 -2 -2

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. 122 140 158 +18 +18 +18 +18 +18 +18 +18

What conjecture can you make about this relationship? For every 10 degrees that Celsius increases, the Fahrenheit increases 18 degrees.

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. 42 The next three terms are _________________, _________________, and ____________________. 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 four person people people people Arrange the information from the simpler problems in a table. Look for a pattern. 45 1 3 6 10 15 21 28 36 1 2 3 4 5 6 7 8 9 Use differences to determine how the number of conversations is increasing. Then extend the pattern to 10 people. 45 With 10 friends, it takes _______ conversations for each person to have exactly one conversation with each other person.