91Ó°ÊÓ

When we begin proving a statement by mathematical induction, the first thing we need to establish is the base case. This step checks if the formula or statement holds true for the very first value, typically when \( n = 1 \). By verifying the base case, we ensure that the pattern we are claiming is on solid footing from the start. In the given example, the base case involves checking if the formula \( 1+7+13+\ldots+(6n-5)=n(3n-2) \) holds when \( n = 1 \). Replacing \( n \) with 1, our formula simplifies to \( 1 = 1 \times (3 \times 1 - 2) \), which is indeed a valid equation since both sides equal 1. Successfully proving the base case means we have a green light to move onto the next stages of induction, laying the foundation for a larger proof.

The inductive step is where we attempt to demonstrate that the truth of our statement for one integer implies its truth for the next integer. This is the pivotal moment in mathematical induction.In our example, we assume that the formula holds for some integer \( k \) and then show that it holds for \( k+1 \). We take the sum up to \((6k-5)\),

• Suppose \( 1+7+13+\ldots+(6k-5) = k(3k-2) \)

Then add the next term, which is \( 6(k+1)-5 \), to both sides. The goal is to demonstrate that including this term results in terms that reflect the formula for \( k+1 \), making it look like

• \((k+1)(3(k+1)-2)\)

Proving this step is crucial as it forms the bridge between theoretical assumption and broader conclusion, thereby showing continuity in the pattern.

The induction hypothesis is our assumption step that is central to the induction process. We assume that our given formula or statement is valid for a certain integer \( k \). This assumption is what will be tested and carried through the inductive step.For example, assume the given formula \( 1+7+13+\ldots+(6 k-5)=k(3 k-2) \) holds true for \( n = k \). This is not the step where we prove anything yet; instead, we set up the logic to show our assumption works for \( k \). By taking this hypothetical truth for \( k \), the induction hypothesis serves as the platform from which we prove the formula holds for \( k+1 \).Thus, the induction hypothesis essentially sets the stage for the inductive process to unfold, allowing us to explore its implications over subsequent numbers.

Algebraic manipulation is vital in mathematical induction, aiding in simplifying and proving various expressions and formulas. It involves rearranging and simplifying expressions to make them more workable or to reveal deeper relationships between terms.During our inductive step, the left-hand side is adjusted by adding \( 6(k+1) - 5 \). This requires algebraic manipulation to show it transforms into the desired form:

• \( k(3k-2) + 6(k+1) - 5 \) becomes \( (k+1)(3(k+1)-2) \)

We perform a series of simplifications and rearrangements to achieve a matching pattern with the right-hand side. This step often involves distribution, combining like terms, and restructuring equations so that both sides equate.Mastering algebraic manipulation empowers students to confidently navigate these transformations, revealing how initial assumptions expand into broader mathematical truths.