Relations & Functions
Functional equations
Grade 12
Question:
<p>It is given that \( f(x+y) = f(x)\,f(y) \) and \( f(1) = 2 \). If \( \displaystyle\sum_{k=1}^{n} f(a+k) = 16(2^n - 1) \), find the value of \( a \).</p>
<p>2</p>
<p>3</p>
<p>4</p>
<p>5</p>
Step-by-Step Solution
Key Concept: Recognize that f(x+y) = f(x)f(y) with f(1) = 2 defines an exponential function f(x) = 2^x. Use this to convert the sum into a geometric series and match coefficients.
<p><strong>Step 1:</strong> Determine f(x) from the functional equation.</p><p>Given f(x+y) = f(x)f(y) and f(1) = 2, this is Cauchy's exponential functional equation.</p><p>For positive integers: f(n) = [f(1)]^n = 2^n</p><p>By extension, f(x) = 2^x for all x.</p><p><strong>Step 2:</strong> Set up the sum using f(x) = 2^x.</p><p>∑(k=1 to n) f(a+k) = ∑(k=1 to n) 2^(a+k) = 2^a · ∑(k=1 to n) 2^k</p><p><strong>Step 3:</strong> Evaluate the geometric series.</p><p>∑(k=1 to n) 2^k = 2(2^n - 1)/(2-1) = 2(2^n - 1)</p><p><strong>Step 4:</strong> Use the given condition.</p><p>2^a · 2(2^n - 1) = 16(2^n - 1)</p><p>2^a · 2 = 16</p><p>2^(a+1) = 2^4</p><p>a + 1 = 4</p><p>∴ Answer: a = 3</p>
Correct Answer: B