Why Does Mathematical Induction Prove Formulas True for All Positive Integers?

[andrewkg]

Hello I'm learning about proofs and in my book there's a sect. On mathematical induction. And I'm trying understand why this makes it true for all values.
1+3+5...2n-1=n^2
Suppose that the formula is known to be true for n=1, and suppose that as a result of assuming that it is true for n=k, where k is an arbitrary positive integer, we can prove that it is also true for n=k+1.
Then the formula is true for all k.

Why does this addition of 1 make it true for all k?

 

[Pagan Harpoon]

You know it's true for n=1 and you know that for every n where it's true, it's also true for n+1. Since you proved it for 1, this implies it's true for 1+1 = 2. Now, since you know it's true for 2, it must be true for 2+1 = 3. Now since you know it's true for 3, it's also true for 3+1 = 4. And so on, so it's true for every positive integer.