Sequences & Series
Mathematical Induction
Grade 11
Question:
<p>For all <span>\(n \in \mathbb{N}\)</span>, <span>\(1 \times 1! + 2 \times 2! + 3 \times 3! + \ldots + n \times n!\)</span> is equal to</p>
<p>(a) <span>\((n+1)! - 2\)</span></p>
<p>(b) <span>\((n+1)!\)</span></p>
<p>(c) <span>\((n+1)! - 1\)</span></p>
<p>(d) <span>\((n+1)! - 3\)</span></p>
Step-by-Step Solution
Key Concept: Use the principle of mathematical induction to prove that the sum of products of factorials equals (n+1)! - 1
<p><strong>Step 1:</strong> Let the statement <span>$P(n)$</span> be defined as</p><p><span>$P(n): 1 \times 1! + 2 \times 2! + 3 \times 3! + \cdots + n \times n! = (n+1)! - 1$</span></p><p>for all natural numbers <span>$n$</span>.</p><p>∴ Answer is (c) <span>$(n+1)! - 1$</span></p>
Correct Answer: c