I'm learning the basics of proof by induction and wanted to see if I took the right steps with the following proof:
Theorem: for $n \in \mathbb{N}, \sum\limits_{k=1}^{n} (2k+1) = n^{2} + 2n $
Base Case:
let $ n = 1 $ Therefore $2*1+1 = 1^{2}+2*1 $ Which proves base case is true.
Inductive Hypothesis:
Assume $\sum_{k=1}^{n} (2k+1) = n^{2} + 2n $
Then $\sum_{k=1}^{n+1} (2k+1) = (n+1)^{2} + 2(n+1) $ $\iff (2(n+1) +1)+ \sum_{k=1}^{n} (2k+1) = (n+1)^{2} + 2(n+1) $ Using inductive hypothesis on summation term: $\iff(2(n+1) +1)+ n^{2} + 2n = (n+1)^{2} + 2(n+1) $ $\iff 2(n+1) = 2(n+1) $
Hence for $n \in \mathbb{N}, \sum\limits_{k=1}^{n} (2k+1) = n^{2} + 2n $ Q.E.D.
Does this prove the theorem? Or was my use of the inductive hypothesis circular logic?