If n is odd, then 3n + 7 is even.
If n is odd, then 3n + 7 is even.
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even.
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove that 3n + 7 is even. Since n is odd, there exists an integer k so that n = 2k + 1.
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove 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
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove 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 + 3 + 7
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove 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 + 3 + 7 3n + 7 = 6k + 10
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove 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 + 3 + 7 3n + 7 = 6k + 10 3n + 7 = 2(3k + 5).
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove 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 + 3 + 7 3n + 7 = 6k + 10 3n + 7 = 2(3k + 5).
Since k is an integer, 3k + 5 is also an integer because integers are closed under addition and multiplication.
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove 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 + 3 + 7 3n + 7 = 6k + 10 3n + 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
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove 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 + 3 + 7 3n + 7 = 6k + 10 3n + 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.
If n is odd, then 3n + 7 is even.
Proof: We suppose that n is an odd integer and want to prove 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 + 3 + 7 3n + 7 = 6k + 10 3n + 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.
Alternative proof:
Proof: We suppose that n is an odd integer and want to prove 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 + 3 + 7 = 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
Writing Guidelines
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.
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.
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.
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)...”
3. 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.”
Writing Guidelines Continued ...
4. 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 $.
