<p>If the third term in the binomial expansion of \((1+x^{\log_2 x})^5\) equals 2560, then a possible value of \(x\) is:</p>
Step-by-Step Solution
Key Concept: Identify the third term using the binomial expansion formula T_{r+1} = C(n,r)a^(n-r)b^r, then set it equal to 2560 and solve for x by recognizing that the exponent of x must form a solvable equation.
<p><strong>Step 1:</strong> Identify the third term in the expansion of (1 + x^(log₂ x))⁵</p><p>The third term corresponds to r = 2, so: T₃ = C(5,2) · 1³ · (x^(log₂ x))²</p><p><strong>Step 2:</strong> Simplify T₃</p><p>T₃ = 10 · x^(2log₂ x) = 10 · x^(log₂ x²)</p><p><strong>Step 3:</strong> Set T₃ = 2560</p><p>10 · x^(log₂ x²) = 2560</p><p>x^(log₂ x²) = 256</p><p><strong>Step 4:</strong> Solve for x by taking log₂ of both sides</p><p>log₂(x^(log₂ x²)) = log₂ 256</p><p>(log₂ x²) · (log₂ x) = 8</p><p>2(log₂ x) · (log₂ x) = 8</p><p>2(log₂ x)² = 8</p><p>(log₂ x)² = 4</p><p>log₂ x = ±2</p><p><strong>Step 5:</strong> Find x</p><p>If log₂ x = 2, then x = 4</p><p>If log₂ x = -2, then x = 1/4</p><p><strong>Verification:</strong> For x = 4: T₃ = 10 · 4^(log₂ 16) = 10 · 4⁴ = 10 · 256 = 2560 ✓</p><p>∴ Answer: x = 4 (or x = 1/4)</p>
Correct Answer: D