Hi all, Im new to the forum and I am having difficulties understanding Inductions.

I have these 2 examples from the textbook and they give me answers but I was wondering if anyone can help me through it. I keep staring at it and it just doesn't sink in.

1st problem:

Prove that the sum of the first N odd positive integers = n^2

I get the first step prove that P(1) = true which is sum of the first odd integer which is 1 and 1^2 = 1.

The textbook puts 1+3+5+...+(2k-1)=k^2

I know 2k-1 ='s an odd integer but this is what confuses me

the textbook has 1+3+5....+(2k-1)+(2k+1) = (k+1)^2

then..

has 1+3+5....+(2k-1)+(2k+1) = [1+3+...(2k-1)] + (2k+1)

=k^2 + (2k+1) ---where did k^2 come from?

=k^2 + 2k + 1

=(k+1)^2