Basic Mathematics & Logarithm
Mathematical Induction
Grade 11
Question:
<p>If <latex>P(n) = 2 + 4 + 6 + \cdots + 2n, n \in \mathbb{N}</latex>, then <latex>P(k) = k(k+1) + 2</latex> and <latex>P(k+1) = (k+1)(k+2) + 2, \forall k \in \mathbb{N}</latex>. So, we can conclude that <latex>P(n) = n(n+1) + 2</latex> for</p>
<p>(a) all <latex>n \in \mathbb{N}</latex></p>
<p>(b) <latex>n \geq 1</latex></p>
<p>(c) <latex>n \geq 2</latex></p>
<p>(d) Nothing can be said</p>
Step-by-Step Solution
Key Concept: Mathematical induction proves a formula for all natural numbers when the inductive step holds for all positive integers.
<p>By mathematical induction, if <latex>P(k)</latex> and <latex>P(k+1)</latex> follow the stated pattern for all <latex>k \in \mathbb{N}</latex>, then <latex>P(n) = n(n+1) + 2</latex> for all natural numbers.</p>
Correct Answer: a