Question

foundations 20 chapter 1 exam name: multiple choice: identify the choice that best completes the statement or answers the question. 1. bill made the following conjecture: when you add a multiple of 6 and a multiple of 9, the sum will be a multiple of 6. is the following equation a counter - example to this conjecture? 12 + 27 = 39 a. yes, it is a counter - example, because 39 is not a multiple of 6. b. yes, it is a counter - example, because 39 is a multiple of 3. c. no, it is not a counter - example, because 39 is a multiple of 3. d. no, it is not a counter - example, because 39 is not a multiple of 9. 2. which conjecture, if any, could you make about the product of two odd integers? a. the product will be an even integer. b. the product will be an odd integer. c. the product will be negative. d. it is not possible to make a conjecture. 3. which of the following choices, if any, uses inductive reasoning to show that the sum of three even integers is even? a. 2x + 2y+2z = 2(x + y + z) b. 2 + 4+6 = 12 and 4 + 6 + 8 = 18 c. x + y+z = 2(x + y + z) d. none of the above choices 4. which of the following choices, if any, uses deductive reasoning to show that the sum of three even integers is even? a. x + y+z = 2(x + y + z) b. 2x + 2y+2z = 2(x + y + z) c. 2 + 4+6 = 12 and 4 + 6 + 8 = 18 d. none of the above choices 5. what type of error, if any, occurs in the following proof? circle the step where the error occurred. 2 = 2 + 2 4(2)=4(2 + 2) 4(2)+3 = 4(2 + 2)+3 8 + 3 = 16 + 3 11 = 19 a. a false premise b. an error in calculation c. there is no error in the proof. 6. determine the unknown term in this pattern. 1, 1, 2, 3, 5, _, 13, 21 a. 6 b. 7 c. 8 d. 9

Explanation:

Step1: Analyze question 1

The conjecture is sum of multiple of 6 and 9 is multiple of 6. 12 is multiple of 6, 27 is multiple of 9, and 39 is not multiple of 6. So it's a counter - example.

Step2: Analyze question 2

Let two odd integers be \(2m + 1\) and \(2n+1\). Their product \((2m + 1)(2n + 1)=4mn+2m + 2n+1=2(2mn + m + n)+1\), which is odd.

Step3: Analyze question 3

Inductive reasoning uses specific examples. \(2 + 4+6 = 12\) and \(4 + 6+8 = 18\) are specific examples to show sum of three even integers is even.

Step4: Analyze question 4

Deductive reasoning uses general forms. \(2x+2y + 2z=2(x + y + z)\) shows sum of three even integers (represented as \(2x\), \(2y\), \(2z\)) is even.

Step5: Analyze question 5

In \(4(2)+3 = 4(2 + 2)+3\), the left - hand side is \(4\times2+3=8 + 3=11\) and the right - hand side is \(4\times4+3=16 + 3 = 19\). The error is in calculation.

Step6: Analyze question 6

This is a Fibonacci - like sequence where each term is sum of previous two terms. \(3+5 = 8\).

Answer:

1. a. Yes, it is a counterexample, because 39 is not a multiple of 6.
2. b. The product will be an odd integer.
3. c. \(2 + 4+6 = 12\) and \(4 + 6+8 = 18\)
4. b. \(2x+2y + 2z=2(x + y + z)\)
5. b. an error in calculation
6. c. 8