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Question 1

1pts

Consider the following example: Prove that for all natural numbersn, 6n- 1 is a multiple of 5. When increasingn, what pattern do you notice with 6n- 1?

Group of answer choices

The last digit is always 1.

The last digit is always 6.

It is always a multiple of 6.

It is always a multiple of 5.

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Question 2

1pts

The statement, "6k- 1 is a multiple of 5" can be written as which of the following equations?

Group of answer choices

6k- 1 = 5

6k- 1 = 5jfor some integerj

5(6k- 1) = 1

6k- 1 = 5 +jfor some integerj

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Question 3

1pts

To prove by induction, we start by saying what the statement is that you want to prove: "LetP(n) be the statement... ." Then, to prove thatP(n) is true for all,n 0, what cases must be proven? (Check all that apply.)

Group of answer choices

Base case.

The case whenn= 1.

Inductive case.

The case whennis any arbitrary number.

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Question 4

1pts

Which of the following must be proven as the inductive case in a proof by induction?

Group of answer choices

P(0)

P(k) for allk 0)

P(k+1) for allk 0

P(k) P(k+1) for allk 0

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Question 5

1pts

Consider the following example: "Prove that for all natural numbersn, 6n- 1 is a multiple of 5."In the final step of induction, we want to show that 6k+1- 1 is also a multiple of 5. So, how can we write 6k+1- 1 in terms of 6k- 1?

Group of answer choices

6k+1- 1 = 6 + 6k- 1

6k+1- 1 = 6 * 6k- 1

6k+1- 1 = 6k

6k+1- 1 = 6k+ 1