Each term is half the preceding term. So the next two terms are

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1-1 Patterns and Inductive Reasoning

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Each term is half the preceding term. So the next two terms are Patterns and Inductive Reasoning LESSON 1-1 Additional Examples Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence. 384, 192, 96, 48, … Each term is half the preceding term. So the next two terms are 48 ÷ 2 = 24 and 24 ÷ 2 = 12. Quick Check

perfect squares form a pattern. Patterns and Inductive Reasoning LESSON 1-1 Additional Examples Make a conjecture about the sum of the cubes of the first 25 counting numbers. Find the first few sums. Notice that each sum is a perfect square and that the perfect squares form a pattern. 13 = 1 = 12 = 12 13 + 23 = 9 = 32 = (1 + 2)2 13 + 23 + 33 = 36 = 62 = (1 + 2 + 3)2 13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2 13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2 The sum of the first two cubes equals the square of the sum of the first two counting numbers.

The sum of the first three cubes equals the square of the sum of the Patterns and Inductive Reasoning LESSON 1-1 Additional Examples (continued) The sum of the first three cubes equals the square of the sum of the first three counting numbers. This pattern continues for the fourth and fifth rows of the table. 13 + 23 + 33 + 43 = 100 = 102 = (1 + 2 + 3 + 4)2 13 + 23 + 33 + 43 + 53 = 225 = 152 = (1 + 2 + 3 + 4 + 5)2 So a conjecture might be that the sum of the cubes of the first 25 counting numbers equals the square of the sum of the first 25 counting numbers, or (1 + 2 + 3 + … + 25)2. Quick Check

Find a counterexample for each conjecture. Patterns and Inductive Reasoning LESSON 1-1 Additional Examples Find a counterexample for each conjecture. a. A number is always greater than its reciprocal. Sample counterexamples: 1 is not greater than = 1. 1 is not greater than 2. 1 2 3 is not greater than – . 1 3 – b. If a number is divisible by 5, then it is divisible by 10. Sample counterexample: 25 is divisible by 5 but not by 10. Quick Check

Write the data in a table. Find a pattern. Patterns and Inductive Reasoning LESSON 1-1 Additional Examples The price of overnight shipping was $8.00 in 2000, $9.50 in 2001, and $11.00 in 2002. Make a conjecture about the price in 2003. Write the data in a table. Find a pattern. 2000 $8.00 2001 2002 $9.50 $11.00 Each year the price increased by $1.50. A possible conjecture is that the price in 2003 will increase by $1.50. If so, the price in 2003 would be $11.00 + $1.50 = $12.50. Quick Check