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

[chris_0101]

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

 

[Dick]

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.

 

[chris_0101]

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.

 

[Dick]

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.

 

[chris_0101]

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.