Prove that the sum of the first n odd numbers equals n squared, using induction

Problem: Prove by mathematical induction that $\sum_{i=1}^n (2i - 1) = 1 + 3 + 5 + \dots + (2n - 1) = n^2$ for all integers $n \ge 1$.

Steps

1. Verify the base case $n = 1$
   $$1 = 1^2$$
2. State the induction hypothesis for $n = k$
   $$1 + 3 + 5 + \dots + (2k - 1) = k^2$$
3. Add the next odd term, $2(k + 1) - 1 = 2k + 1$, to both sides
   $$1 + 3 + 5 + \dots + (2k - 1) + (2k + 1) = k^2 + 2k + 1$$
4. Factor the right-hand side
   $$1 + 3 + 5 + \dots + (2k + 1) = (k + 1)^2$$