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The inductive hypothesis is a crucial part of mathematical induction, a technique used to prove that a statement is true for all positive integers. This concept involves assuming that a statement is true for a particular positive integer, often denoted as \(k\).
We use this assumption to solve a step further. The inductive hypothesis for our example assumes that the formula \(2 + 7 + 12 + \cdots + (5k - 3) = \frac{1}{2} k(5k - 1)\) holds true for some integer \(k\). This assumption acts as a bridge to prove the next step.
By assuming the statement is true for \(k\), we can investigate whether this supposition allows the statement to also hold for \(k+1\), thereby reinforcing the pattern or formula of the entire sequence. This hypothesis serves as the foundation for completing the proof that the statement applies to all subsequent integers.

The base case is the starting point of any induction proof. It establishes the initial truth of the statement for the smallest positive integer in your series, commonly \(n = 1\).
To validate our statement at the base case: for \(n = 1\), the left-hand side evaluates to \(2\), since \(5 \times 1 - 3 = 2\).
The right-hand side also simplifies to \(2\), as \(\frac{1}{2} \times 1 \times (5 \times 1 - 1) = 2\).

• Verify by substituting: This simple test ensures that at the beginning of the sequence, both expressions equate to the same result.

Verifying the base case confirms that our starting point holds true, laying the groundwork for the subsequent step-by-step induction process.

The inductive step is where we show that if the statement holds for some positive integer \(k\), then it must hold for \(k+1\). Here, inductive reasoning is fully in action.
For our problem, once the statement is assumed true for \(k\), we need to demonstrate it for \(k+1\). Begin by taking the expression \((2 + 7 + 12 + \cdots + (5k - 3)) + (5(k+1) - 3)\).
Using the inductive hypothesis, substitute in the known formula for the sequence up to \(k\), then simplify and combine:

• Replace terms: First, implement the step using the inductive assumption into your current equation.
• Simplify: Work to reduce it down to show how it will hold for \(k+1\).

This step involves algebraic manipulation to demonstrate the next iteration, ensuring that the initial inductive hypothesis is what enables this future proofing.

Algebraic proof in the context of mathematical induction involves a careful blend of hypothesis and algebra skills to establish a statement's validity across a sequence. This typically includes steps where you:
1. Substitute: Start from where left off—the inductive hypothesis provides a known form.
2. Simplify: Perform algebraic manipulation like combining terms and expanding brackets.
3. Compare: Each term should align with the desired form for \(k+1\).
You must ensure that terms are combined correctly and transform the expression into the exact form of the right side for \(n = k+1\).
This technique concludes with showing that our simplified form and the expected formula for \(k+1\) are identical, thereby completing the proof that the statement holds for all \(n\), fulfilling our algebraic proof requirements.