How to show a proof easily - prove that 2k - 1 is odd

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In summary: This suggests to me that this is the correct answer.In summary, if 2k-1 is odd, then 2k-1=2k+1. This can be proven using the definition of odd for any integer k.
  • #1
chris_0101
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Homework Statement


Simply put: prove that 2k - 1 is odd
I am having difficulty trying to show the above proof. If anyone can show me how to complete this, that would be great. Also if anyone can provide tips on the best way to prove statements such as these that would also be appreciated


Homework Equations


n/a


The Attempt at a Solution



if 2k - 1 is odd, then 2k - 1 = 2k + 1, by the definition of odd for any integer k.

This is what I did:
2k - 1 = 2k + 1
-1 = 1 --> but this is wrong and to be honest a really dumb answer, sorry.

Thanks
 
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  • #2
Well, knowing an answer is dumb is good first step. The definition of odd is that an integer n can be written in the form n=2k+1 for SOME integer k. 2k-1 can also be written in the form 2*(some integer)+1. That (some integer) isn't k, but is an integer related to k. What might it be? Don't try to label all possibly different integers in your proof with the same label k.
 
  • #3
Okay I did established the first part of your reply regarding the definition of an odd integer. But the main problem I am having is that how do you actually determine the value that is related to k. I know that I am cheating on this by looking at the back of the book for the answer, but that unknown integer is k -1, so

2(k-1) + 1 = 2k - 1, therefore making it odd

Now, I would of never thought of k - 1 being the answer, why isn't the answer k - 2 or k -3 and so fourth. How to I know when to draw the line and actually determine that, in this case, that k-1 is the answer.
 
  • #4
chris_0101 said:
Okay I did established the first part of your reply regarding the definition of an odd integer. But the main problem I am having is that how do you actually determine the value that is related to k. I know that I am cheating on this by looking at the back of the book for the answer, but that unknown integer is k -1, so

2(k-1) + 1 = 2k - 1, therefore making it odd

Now, I would of never thought of k - 1 being the answer, why isn't the answer k - 2 or k -3 and so fourth. How to I know when to draw the line and actually determine that, in this case, that k-1 is the answer.

You want to prove 2k-1 is odd. That means you want to show 2k-1=2j+1 for some integer j. Notice how I used a different symbol for (some integer) instead of k. Because (some integer) in one expression doesn't have to be the same as (some integer) in the other expression. Now solve for j in terms of k.
 
  • #5
Thank you for your help. I managed to get the same answer, I also tried the same method for other questions and received the same result.
 

1. How do I start a proof for 2k-1 being odd?

To start a proof for 2k-1 being odd, you can assume that k is any integer and use the definition of an odd number, which is a number that can be expressed as 2n+1, where n is any integer.

2. What is the general approach for proving that 2k-1 is odd?

The general approach for proving that 2k-1 is odd is to manipulate the expression 2k-1 to show that it can be rewritten as 2n+1, where n is an integer. This will prove that 2k-1 is an odd number.

3. Can I use mathematical induction to prove that 2k-1 is odd?

Yes, mathematical induction is a valid method for proving that 2k-1 is odd. You can use the base case of k=1 to show that 2(1)-1=1, which is an odd number. Then, you can use the inductive hypothesis to show that if 2k-1 is odd, then 2(k+1)-1 is also odd. This will prove that 2k-1 is odd for all integers k.

4. Are there any other methods for proving that 2k-1 is odd?

Yes, there are other methods for proving that 2k-1 is odd. You can use a direct proof by substituting any integer value for k into the expression 2k-1 and showing that the result is an odd number. You can also use a proof by contradiction by assuming that 2k-1 is even and showing that it leads to a contradiction.

5. Is it necessary to prove that 2k-1 is odd or can it be assumed?

In mathematical proofs, it is always necessary to provide a proof for any statement or claim. Therefore, it is necessary to prove that 2k-1 is odd, rather than assuming it to be true. This ensures the validity and credibility of the proof.

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