Use mathematical induction to prove that each statement is true for every positive integer n. 1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) = n(n + 1)(n + 2)/3

검증된 단계별 안내

Step 1: Define the statement to prove by induction. Let \( P(n) \) be the statement \( 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \cdots + n(n + 1) = \frac{n(n + 1)(n + 2)}{3} \). We want to prove \( P(n) \) is true for all positive integers \( n \).

Step 2: Verify the base case \( n = 1 \). Substitute \( n = 1 \) into both sides of the equation and check if the left-hand side equals the right-hand side.

Step 3: Assume the induction hypothesis \( P(k) \) is true for some positive integer \( k \), that is, assume \( 1 \cdot 2 + 2 \cdot 3 + \cdots + k(k + 1) = \frac{k(k + 1)(k + 2)}{3} \).

Step 4: Prove the statement \( P(k + 1) \) is true using the induction hypothesis. Start with the left-hand side of \( P(k + 1) \): \( 1 \cdot 2 + 2 \cdot 3 + \cdots + k(k + 1) + (k + 1)(k + 2) \). Substitute the induction hypothesis for the sum up to \( k(k + 1) \).

Step 5: Simplify the expression obtained in Step 4 and show that it equals the right-hand side of \( P(k + 1) \), which is \( \frac{(k + 1)(k + 2)(k + 3)}{3} \). This completes the induction step and proves the statement for all positive integers \( n \).

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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.

Summation of Sequences

Summation of sequences involves finding a formula that represents the sum of terms in a sequence. In this problem, the sum is of products of consecutive integers, and the goal is to express this sum in a closed form, which simplifies calculations and helps in proving the formula by induction.

Algebraic Manipulation

Algebraic manipulation refers to the process of simplifying, expanding, or factoring expressions to transform them into a desired form. In induction proofs, it is essential to manipulate expressions correctly to show that the formula for n+1 follows logically from the formula for n.