<p>The middle term in the expansion of \((x/2 + 2)^8\) is 1120; then \(x \in \mathbb{R}\) is equal to</p>
Step-by-Step Solution
Key Concept: In the expansion of (x/2 + 2)^8, identify which term is 'middle' (the 5th term), apply the binomial coefficient formula, and equate it to 1120 to solve for x.
<p><strong>Step 1:</strong> For (x/2 + 2)^8, there are 9 terms (r = 0 to 8). The middle term is T₅ (when r = 4).</p><p><strong>Step 2:</strong> General term: T_{r+1} = C(8,r)(x/2)^{8-r}(2)^r</p><p><strong>Step 3:</strong> For T₅ (r = 4): T₅ = C(8,4)(x/2)^4(2)^4 = 70 · (x⁴/256) · 16</p><p><strong>Step 4:</strong> T₅ = 70 · x⁴ · 16/256 = 70 · x⁴/16 = 35x⁴/8</p><p><strong>Step 5:</strong> Equate to 1120: (35x⁴)/8 = 1120</p><p><strong>Step 6:</strong> 35x⁴ = 8960 → x⁴ = 256 → x⁴ = 2⁸</p><p><strong>Step 7:</strong> x = ±4</p><p>∴ Answer: AD (x = 4 or x = -4)</p>
Correct Answer: AD