1. (For help, go the Skills Handbook, page 715.) GEOMETRY LESSON 1-1 1. Make a list of the positive even numbers. 2. Make a list of the positive odd numbers. 3. Copy and extend this list to show the first 10 perfect squares. 1 2 = 1, 2 2 = 4, 3 2 = 9, 4 2 = 16,... 4. Which do you think describes the square of any odd number? It is odd. It is even. Patterns and Inductive Reasoning 1-1 Here is a list of the counting numbers: 1, 2, 3, 4, 5,... Some are even and some are odd.

2. 1. Even numbers end in 0, 2, 4, 6, or 8: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24,... 2. Odd numbers end in 1, 3, 5, 7, or 9: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25,... 3. 1 2 = (1)(1) = 1; 2 2 = (2)(2) = 4; 3 2 = (3)(3) = 9; 4 2 = (4)(4) = 16; 5 2 = (5)(5) = 25; 6 2 = (6)(6) = 36; 7 2 = (7)(7) = 49; 8 2 = (8)(8) = 64; 9 2 = (9)(9) = 81; 10 2 = (10)(10) = 100 4. The odd squares in Exercise 3 are all odd, so the square of any odd number is odd. Solutions GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

3. -Reasoning based on patterns you observe -Creating logical generalizations -Reasoning from detailed facts to general principles Inductive Reasoning GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

4. Find a pattern for the sequence. Use the pattern to show the next two terms in the sequence. 384, 192, 96, 48, … GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

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

6. A conclusion you reach using inductive reasoning Conjecture GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

7. 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. 1 3 = 1= 1 2 = 1 2 1 3 + 2 3 = 9= 3 2 = (1 + 2) 2 1 3 + 2 3 + 3 3 = 36= 6 2 = (1 + 2 + 3) 2 1 3 + 2 3 + 3 3 + 4 3 = 100= 10 2 = (1 + 2 + 3 + 4) 2 1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 225= 15 2 = (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. GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

8. This pattern continues for the fourth and fifth rows of the table. 1 3 + 2 3 + 3 3 + 4 3 = 100= 10 2 = (1 + 2 + 3 + 4) 2 1 3 + 2 3 + 3 3 + 4 3 + 5 3 = 225= 15 2 = (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. The sum of the first three cubes equals the square of the sum of the first three counting numbers. (continued) GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

9. A single example that proves a conjecture to be false. Counter Example GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 Definitions

10. The first three odd prime numbers are 3, 5, and 7. Make and test a conjecture about the fourth odd prime number. The fourth prime number is 11. One pattern of the sequence is that each term equals the preceding term plus 2. So a possible conjecture is that the fourth prime number is 7 + 2 = 9. However, because 3 X 3 = 9 and 9 is not a prime number, this conjecture is false. GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

11. 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 20012002 $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. GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

12. GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning Pages 6–9 Exercises 1.80, 160 2.33,333; 333,333 3.–3, 4 4., 5.3, 0 6.1, 7.N, T 8.J, J 9.720, 5040 10.64, 128 11., 1 16 1 32 1 36 1 49 12., 13.James, John 14.Elizabeth, Louisa 15.Andrew, Ulysses 16.Gemini, Cancer 17. 18. 1515 1616 19.The sum of the first 6 pos. even numbers is 6 7, or 42. 20.The sum of the first 30 pos. even numbers is 30 31, or 930. 21.The sum of the first 100 pos. even numbers is 100 101, or 10,100. 1313 1-1

13. 28. ÷ = and is improper. 29.75°F 30.40 push-ups; answers may vary. Sample: Not very confident, Dino may reach a limit to the number of push-ups he can do in his allotted time for exercises. 31.31, 43 32.10, 13 33.0.0001, 0.00001 34.201, 202 35.63, 127 36., 37.J, S 38.CA, CO 39.B, C 1313 1212 1313 1313 1212 1212 // / 1212 1313 3232 3232 31 32 63 64 GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1 22.The sum of the first 100 odd numbers is 100 2, or 10,000. 23.555,555,555 24.123,454,321 25–28. Answers may vary. Samples are given. 25.8 + (–5 = 3) and 3 > 8 26. > and > 27.–6 – (–4) < –6 and –6 – (–4) < –4

14. 40.Answers may vary. Sample: In Exercise 31, each number increases by increasing multiples of 2. In Exercise 33, to get the next term, divide by 10. 41. You would get a third line between and parallel to the first two lines. 42. 43. 44. 45. 46.102 cm 47.Answers may vary. Samples are given. a. Women may soon outrun men in running competitions. b. The conclusion was based on continuing the trend shown in past records. c. The conclusions are based on fairly recent records for women, and those rates of improvement may not continue. The conclusion about the marathon is most suspect because records date only from 1955. GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

15. 50.His conjecture is probably false because most people’s growth slows by 18 until they stop growing somewhere between 18 and 22 years. 51.a. b.H and I c.a circle 48.a. b.about 12,000 radio stations in 2010 c.Answers may vary. Sample: Confident; the pattern has held for several decades. 49.Answers may vary. Sample: 1, 3, 9, 27, 81,... 1, 3, 5, 7, 9,... 52.21, 34, 55 53.a.Leap years are years that are divisible by 4. b.2020, 2100, and 2400 c.Leap years are years divisible by 4, except the final year of a century which must be divisible by 400. So, 2100 will not be a leap year, but 2400 will be. GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

16. 54.Answers may vary. Sample: 100 + 99 + 98 + … + 3 + 2 + 1 1 + 2 + 3 + … + 98 + 99 + 100 101 + 101 + 101 + … + 101 + 101 + 101 The sum of the first 100 numbers is, or 5050. The sum of the first n numbers is. 55.a.1, 3, 6, 10, 15, 21 b.They are the same. c.The diagram shows the product of n and n + 1 divided by 2 when n = 3. The result is 6. 100 101 2 n(n+1) 2 55. (continued) d. 56.B 57.I 58.[2] a. 25, 36, 49 b. n 2 [1] one part correct GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1

17. 59.[4] a.The product of 11 and a three-digit number that begins and ends in 1 is a four-digit number that begins and ends in 1 and has middle digits that are each one greater than the middle digit of the three-digit number. (151)(11) = 1661 (161)(11) = 1771 b. 1991 c. No; (191)(11) = 2101 59. (continued) [3]minor error in explanation [2]incorrect description in part (a) [1]correct products for (151)(11), (161)(11), and (181)(11) 60-67. 68.B 69.N 70.G GEOMETRY LESSON 1-1 Patterns and Inductive Reasoning 1-1