Proof of 1 + 3 + 5 + ... + (2n -1) = n2

[zeion]

Homework Statement

Show that the statement holds for all positive integers n.

1 + 3 + 5 + ... + (2n -1) = n²

Homework Equations

The Attempt at a Solution

Assume that k will work, then k + 1:

1 + 3 + 5 + ... + (2(k+1) -1) = (k+1)²
1 + 3 + 5 + ... + 2k+1 = k² + 2k + 1

Recall that for k,
1 + 3 + 5 + ... + (2k -1) = k²

Then k+1,
1 + 3 + 5 + ... + (2k -1) + (2k+1) = k² + (2k + 1)

Is this enough to conclude that the statement holds?

 

[n!kofeyn]

Yes, that is correct. I would just be a little careful in your wording and progression of statements. I know that you are working backwards so to speak, but from appearances, it looks like you are assuming what you are trying to prove. When doing induction, try to be as strict and algorithmic as possible. Make sure each statement follows directly from the one preceding it.

 

[icystrike]

Yes, that is correct. I would just be a little careful in your wording and progression of statements. I know that you are working backwards so to speak, but from appearances, it looks like you are assuming what you are trying to prove. When doing induction, try to be as strict and algorithmic as possible. Make sure each statement follows directly from the one preceding it.

I guess you ought to mention the equation satisfy for n=1 .
This is important as 1 is the first pst integer.