<p>Find the second term after the middle term in the expansion of \(\left(\dfrac{x}{2} + \dfrac{2}{x^2}\right)^{20}\).</p>
Step-by-Step Solution
Key Concept: In a binomial expansion with even power 2n, the middle term is the (n+1)th term. The 'second term after middle' means we need the (n+2)th term. For (a+b)^20, the middle term is the 11th term, so we need the 13th term, which corresponds to r=12 in the general term T_{r+1}.
<p><strong>Step 1:</strong> Identify the middle term position. For (a+b)^20, there are 21 terms total (r = 0 to 20). The middle term is the 11th term (when r=10). The second term after the middle term is the 13th term, which corresponds to r=12.</p><p><strong>Step 2:</strong> Use the general term formula: T_{r+1} = C(20,r) · (x/2)^{20-r} · (2/x²)^r</p><p><strong>Step 3:</strong> Substitute r=12:</p><p>T_{13} = C(20,12) · (x/2)^8 · (2/x²)^{12}</p><p><strong>Step 4:</strong> Simplify the powers:</p><p>T_{13} = C(20,12) · (x^8/2^8) · (2^{12}/x^{24})</p><p>T_{13} = C(20,12) · (2^{12}/2^8) · (x^8/x^{24})</p><p>T_{13} = C(20,12) · 2^4 · x^{-16}</p><p><strong>Step 5:</strong> Calculate C(20,12) = C(20,8) = 125970</p><p>T_{13} = 125970 · 16 · x^{-16} = <strong>2015520/x^{16}</strong> or <strong>2015520 · x^{-16}</strong></p><p>∴ Answer: <strong>2015520 · x^{-16}</strong> (or 2015520/x^16)</p>
Correct Answer: 2015520