Sets, Relations & Functions
General
Grade 11
Question:
Let <span class="math-inline">\(\sum_{k=1}^{10} f(a + k) = 16(2^{10} - 1)\)</span>, where the function f satisfies <span class="math-inline">\(f(x + y) = f(x)f(y)\)</span> for all natural numbers <span class="math-inline">\(x, y\)</span> and <span class="math-inline">\(f(1) = 2\)</span>. then the natural number 'a' is
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f(k)=2k. g(n)=Σ2k=n(n-1). n(n-1)=20 → n=5.</p><div class="key-concept"><strong>Key Concept:</strong> Additive f with f(1)=c → f(n)=cn on ℕ</div></div>
Correct Answer: 2