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Using Differences to Identify Patterns Section 1.1 number _________ sequence is a string of • A _______ numbers, or terms, in a certain order. • If the difference from one term to the next in a number sequence is always the same the difference is called a ______ constant ___________. difference _________ Example 1: • Find the next three terms of each sequence by using constant differences. A. 1, 3, 5, 7, 9, … 11 1 3 5 7 9 ___ +2 +2 +2 +2 +2 13 ___ 15 ___ +2 +2 Example 1 B. 80, 73, 66, 59, 52, … • 80 73 66 59 52 ___ 45 -7 -7 -7 -7 -7 ___ 38 ___ 31 -7 -7 Try these… C. 1, 4, 7, 10, 13, … 1 4 7 10 13 ___ ___ ___ 16 19 22 +3 +3 +3 +3 +3 +3 +3 Try these… D. 30, 25, 20, 15, 10, … • 0 ___ 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. E. 1, 4, 9,16, 25, … 1 4 9 16 25 ___ 36 ___ 49 ___ 64 +3 +5 +2 +7 +2 +9 +2 +11 +2 +13 +2 +15 +2 First differences Second differences Example 2 • F. 37, 41, 48, 58, 71, … • 87 106 128 37 41 48 58 71 +4 +7 +3 +10 +13 +3 +3 +16 +3 +19 +3 +22 +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, … 42 ___ 56 72 2 6 12 20 30 ___ ___ +4 +6 +2 +8 +2 +10 +2 +12 +2 +14 +2 +16 +2 First differences Second differences 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 +12 +10 -2 +8 -2 +6 -2 +4 -2 +2 -2 +0 -2 First differences Second differences • A___________ conjecture 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. 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 86 104 122 140 158 +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. 12 2 23 6 34 12 45 20 6 7 42 The next three terms are _________________, 7 8 56 8 9 72 _________________, and ____________________. 56 30 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 person two people three people four people Arrange the information from the simpler problems in a table. Look for a pattern. People Conversations 1 2 3 4 5 6 7 8 9 10 0 1 1 10 15 21 28 36 45 3 6 2 3 4 5 6 7 8 9 Use differences to determine how the number of conversations 45 conversations With 10 friends, it takes _______ for each person to have exactly is increasing. Then extend the pattern to 10 people. one conversation with each other person.