Sets, Relations & Functions
General
Grade 11
Question:
<p>Let <span class="math-inline">\(\sum_{k=1}^{10}f(a+k)=16(2^{10}-1)\)</span>, <span class="math-inline">\(f(x+y)=f(x)f(y)\)</span>, <span class="math-inline">\(f(1)=2\)</span>. Find <span class="math-inline">\(a\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f(n)=2ⁿ. Sum=2ᵃ·Σ2ᵏ(k=1..10)=2ᵃ·2(2¹⁰-1)=16(2¹⁰-1) → 2ᵃ⁺¹=16=2⁴ → a=3.</p><p><strong>Answer: a=3</strong></p><div class="trap-box"><strong>Trap:</strong> f(x+y)=f(x)f(y) is multiplicative-on-addition, not f(xy).</div><div class="key-concept"><strong>Key Concept:</strong> Additive-input multiplicative f → f(n)=f(1)ⁿ</div></div>
Correct Answer: 3