If n is odd, then 3n + 7 is even.

Transcription

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2 Proof:

3 Proof: We suppose...

4 that 3n + 7 is even.

5 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1.

6 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7

7 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k

8 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k n + 7 = 6k + 10

9 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k n + 7 = 6k n + 7 = 2(3k + 5).

10 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k n + 7 = 6k n + 7 = 2(3k + 5). Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication.

11 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k n + 7 = 6k n + 7 = 2(3k + 5). Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication. If we let q be the integer 3k + 5, then by substitution we have shown 3n + 7 = 2q.

12 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k n + 7 = 6k n + 7 = 2(3k + 5). Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication. If we let q be the integer 3k + 5, then by substitution we have shown 3n + 7 = 2q. Therefore, if n is an odd integer, we have shown that 3n + 7 is even.

13 that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 3n + 7 = 6k n + 7 = 6k n + 7 = 2(3k + 5). Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication. If we let q be the integer 3k + 5, then by substitution we have shown 3n + 7 = 2q. Therefore, if n is an odd integer, we have shown that 3n + 7 is even.

14 Alternative proof: that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1. By substituting for n and using algebra, we get 3n + 7 = 3(2k + 1) + 7 = 6k = 6k + 10 = 2(3k + 5) = 2q, where q = 3k +5 is an integer because k is an integer, and integers are closed under addition and multiplication. Therefore, we have shown that 3n + 7 is even when n is an odd integer. QED

15 Writing Guidelines We do not consider a proof complete until there is a well-written proof.

16 Writing Guidelines We do not consider a proof complete until there is a well-written proof. 0. Do all of the thinking, work, and planning first. A Know-Show table or outline or notes or scratch work should be completed prior to writing so you can focus on quality writing, not math.

17 Writing Guidelines We do not consider a proof complete until there is a well-written proof. 0. Do all of the thinking, work, and planning first. A Know-Show table or outline or notes or scratch work should be completed prior to writing so you can focus on quality writing, not math. 1. Begin with a carefully worded statement of the theorem or result to be proven. State what you are about to prove. Below that write Proof and begin writing.

18 Writing Guidelines We do not consider a proof complete until there is a well-written proof. 0. Do all of the thinking, work, and planning first. A Know-Show table or outline or notes or scratch work should be completed prior to writing so you can focus on quality writing, not math. 1. Begin with a carefully worded statement of the theorem or result to be proven. State what you are about to prove. Below that write Proof and begin writing. 2. Begin the proof with a statement of assumptions. We assume (the hypothesis)... or Suppose (the hypothesis)...

19 Writing Guidelines We do not consider a proof complete until there is a well-written proof. 0. Do all of the thinking, work, and planning first. A Know-Show table or outline or notes or scratch work should be completed prior to writing so you can focus on quality writing, not math. 1. Begin with a carefully worded statement of the theorem or result to be proven. State what you are about to prove. Below that write Proof and begin writing. 2. Begin the proof with a statement of assumptions. We assume (the hypothesis)... or Suppose (the hypothesis) Use the pronoun we. Mathematicians are a loving community that does everything together. Do not use I, my, you or similar pronouns in writing proofs. It is our convention that we use the pronouns we and our and us.

20 Writing Guidelines Continued Use italics for variables when typing. 5. Display important equations and mathematical expressions. They should be centered and well-aligned. 6. Tell the reader when you are done. Give some form of QED: Quod Erat Demonstrandum - which was to be demonstrated. Use whatever symbol you like: or or or or $.