Question 29- How to conclude the following proof.

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    Proof
Since $l$ is an integer, so is $l^2$ and $l^2+ l$ and $k$. That proves that $n^2= 4k+ 1$ for some integer $k$.
  • #1
cbarker1
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Dear Everyone,
I have trouble writing the conclusion of the proof.
29. The square of every odd integer is one more than an integral multiple of 4.
Work:
Let $n\in\Bbb{Z}$
If n is odd, then $n^2=1+4k$ for some $k\in\Bbb{Z}$.

Examples
Let n=3. Then k=2.
Let n=5. Then k=6.
Let n=21. Then k=110.

Proof:
Suppose n is odd. Then $n=1+4l$ for some $l\in\Bbb{Z}$.
Then, $(2l+1)^2=1+4k$
$4l^2+4l+1=1+4k$
$4(l^2+l)+1=1+4k$
Since $4(l^2+l)\in\Bbb{Z}$. Then...

Thanks for the Help,
CBarker1
 
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  • #2
If $n$ is an odd integer then it is of the form $2k-1$. Squaring, we find $(2k-1)^2=4k^2-4k+1=4(k^2-k)+1$ hence $n^2$ is one more than an integral multiple of $4$.
 
  • #3
Cbarker1 said:
Dear Everyone,
I have trouble writing the conclusion of the proof.
29. The square of every odd integer is one more than an integral multiple of 4.
Work:
Let $n\in\Bbb{Z}$
If n is odd, then $n^2=1+4k$ for some $k\in\Bbb{Z}$.

Examples
Let n=3. Then k=2.
Let n=5. Then k=6.
Let n=21. Then k=110.

Proof:
Suppose n is odd. Then $n=1+4l$ for some $l\in\Bbb{Z}$.
No. This is not true for n= 3 or 7 or 11, etc. What is true that $n= 1+ 2l$.

Then, $(2l+1)^2=1+4k$
Okay, so the "4" before was a typo. Now, this is what you want to prove.

$4l^2+4l+1=1+4k$
$4(l^2+l)+1=1+4k$
Since $4(l^2+l)\in\Bbb{Z}$. Then...

Thanks for the Help,
CBarker1
Not quite a valid proof because in stating "$(2l+ 1)^2= 4k+ 1$" you are assuming what you want to prove.

Instead, starting with n is odd, so $n= 2l+ 1$, we have $n^2= (2l+ 1)^2= 4l^2+ 4l+ 1= 4(l^2+ l)+ 1= 4k+ 1$ where $k= l^2+ l$.
 

FAQ: Question 29- How to conclude the following proof.

1. How do I know when to conclude a proof?

There is no set rule for when to conclude a proof, as it depends on the specific problem and proof structure. However, a good indicator is when you have reached the desired result or proven the statement you set out to prove.

2. What should be included in a conclusion of a proof?

A conclusion should summarize the main points of the proof and restate the desired result. It should also explain how the result was reached and any important implications or applications of the proof.

3. Can I use informal language in the conclusion of a proof?

It is best to use formal language in a proof, including the conclusion. This helps to clearly and precisely convey the logic and reasoning behind the proof.

4. Is it necessary to include a concluding statement in a proof?

Yes, a concluding statement is an important part of a proof as it ties together all the previous steps and clearly states the result that has been proven. It also helps to make the proof more organized and easy to follow.

5. Are there any common mistakes to avoid when concluding a proof?

One common mistake is to introduce new information or assumptions in the conclusion that were not previously stated or proven in the proof. It is important to only use information and assumptions that have been previously established in the proof.

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