Use mathematical induction to prove that each statement is true for every positive integer n. 4 + 8 + 12 + ... + 4n = 2n(n + 1)

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Identify the statement to prove using mathematical induction: For every positive integer \(n\), the sum \(4 + 8 + 12 + \ldots + 4n\) equals \(2n(n + 1)\).

Base Case: Verify the statement for \(n = 1\). Substitute \(n = 1\) into both sides of the equation and check if they are equal.

Inductive Hypothesis: Assume the statement is true for some positive integer \(k\), that is, assume \(4 + 8 + 12 + \ldots + 4k = 2k(k + 1)\) holds.

Inductive Step: Using the inductive hypothesis, prove the statement is true for \(k + 1\). Start with the left side for \(n = k + 1\): \(4 + 8 + 12 + \ldots + 4k + 4(k + 1)\).

Show that adding \(4(k + 1)\) to the sum \(2k(k + 1)\) (from the inductive hypothesis) simplifies to \(2(k + 1)((k + 1) + 1)\), which matches the right side of the formula for \(n = k + 1\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers. It involves two steps: proving the base case (usually n=1) is true, and then proving that if the statement holds for an arbitrary integer k, it also holds for k+1. This creates a chain of truth for all n.

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence, where each term increases by a constant difference. In this problem, the series 4 + 8 + 12 + ... + 4n has a common difference of 4. Understanding how to express and sum such series is essential to verify the formula given.

Formula for the Sum of an Arithmetic Series

The sum of the first n terms of an arithmetic series can be calculated using the formula S_n = n/2 (first term + last term). Applying this formula helps to derive or verify the closed-form expression 2n(n + 1) for the given series, which is crucial for the induction proof.