Sets, Relations & Functions
Functional Equations
Grade 11
Question:
<p>We have \(f(x + y) = f(x) + f(y)\) for all \(x, y \in \mathbb{R}\), \(f(0) = 0\), \(f(1) = 7\). Then \(\displaystyle\sum_{r=1}^{n} f(r)\) equals:</p>
<p>(A) \(\dfrac{7n(n+1)}{2}\)</p>
<p>(B) \(7n(n+1)\)</p>
<p>(C) \(\dfrac{7n}{2}\)</p>
<p>(D) \(7(n+1)\)</p>
Step-by-Step Solution
Key Concept: Use Cauchy's functional equation to determine f(x) = 7x, then apply the arithmetic sum formula to find the telescoping series.
<p><strong>Step 1:</strong> Use the functional equation f(x+y) = f(x) + f(y) (Cauchy's equation).</p><p>Setting y=1 repeatedly: f(n) = f(1+1+...+1) = nf(1) = 7n for all positive integers n.</p><p><strong>Step 2:</strong> Calculate the sum:</p><p>∑(r=1 to n) f(r) = ∑(r=1 to n) 7r = 7∑(r=1 to n) r = 7 · n(n+1)/2</p><p><strong>Step 3:</strong> Verify: f(0)=0 ✓, f(1)=7 ✓, and the functional equation is satisfied since f(x)=7x is additive.</p><p>∴ Answer: <strong>7n(n+1)/2</strong></p>
Correct Answer: A