1. **Problem:** Prove by mathematical induction that for every positive integer $n$, $$5 + 7 + 9 + \ldots + (2n + 3) = n(n + 4)$$ 2. **Base Case:** For $n=1$, the left side is $5$ and the right side is $1(1+4) = 5$. Both sides are equal, so the base case holds. 3. **Inductive Hypothesis:** Assume the formula holds for some positive integer $k$, i.e., $$5 + 7 + 9 + \ldots + (2k + 3) = k(k + 4)$$ 4. **Inductive Step:** We need to prove the formula holds for $k+1$: $$5 + 7 + 9 + \ldots + (2k + 3) + [2(k+1) + 3] = (k+1)(k+5)$$ Using the inductive hypothesis, substitute the sum up to $k$: $$k(k + 4) + [2(k+1) + 3] = (k+1)(k+5)$$ Simplify the added term: $$k(k + 4) + (2k + 2 + 3) = k(k + 4) + (2k + 5)$$ Expand and simplify the left side: $$k^2 + 4k + 2k + 5 = k^2 + 6k + 5$$ Expand the right side: $$(k+1)(k+5) = k^2 + 5k + k + 5 = k^2 + 6k + 5$$ Both sides are equal, so the formula holds for $k+1$. 5. **Conclusion:** By mathematical induction, the formula $$5 + 7 + 9 + \ldots + (2n + 3) = n(n + 4)$$ is true for all positive integers $n$.