Sets, Relations & Functions
General
Grade 11
Question:
<p>Let <span class="math-inline">\(f(x)=\dfrac{4^x}{4^x+2}\)</span> on <span class="math-inline">\([0,1]\)</span>. Find <span class="math-inline">\(\displaystyle\sum_{k=1}^{39}f\!\left(\tfrac{k}{40}\right)-f\!\left(\tfrac{1}{2}\right)\)</span>.</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f(x)+f(1-x)=1 (verify: 4^x/(4^x+2)+4^{1-x}/(4^{1-x}+2)=1). Pair k and 40-k: 19 pairs each summing to 1. Plus middle term f(20/40)=f(1/2). Total sum=19+f(1/2). Subtract f(1/2): answer=19.</p><p><strong>Answer: 19</strong></p><div class="trap-box"><strong>Trap:</strong> Don't compute terms individually. Pairing identity is the key.</div><div class="key-concept"><strong>Key Concept:</strong> f(x)+f(1-x)=1 pairing identity for symmetric sums</div></div>
Correct Answer: 19