<p>Find \(n\), if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of \(\left(\sqrt[4]{2} + \dfrac{1}{\sqrt[4]{3}}\right)^n\) is \(\sqrt{6} : 1\).</p>
Step-by-Step Solution
Key Concept: The fifth term from the end in a binomial expansion of (a+b)^n is the (n-4)th term from the beginning. Use the ratio of general terms and exploit the symmetry of binomial coefficients: C(n,r) = C(n,n-r).
<p><strong>Step 1:</strong> Identify the terms. Fifth term from beginning is T₅ = C(n,4)·(2^(1/4))^(n-4)·(3^(-1/4))⁴. Fifth term from end is T_(n-4) = C(n,n-4)·(2^(1/4))⁴·(3^(-1/4))^(n-4).</p><p><strong>Step 2:</strong> Since C(n,4) = C(n,n-4), the ratio becomes:</p><p>T₅/T_(n-4) = [(2^(1/4))^(n-4)·(3^(-1/4))⁴]/[(2^(1/4))⁴·(3^(-1/4))^(n-4)]</p><p>= (2^(1/4))^(n-8)·(3^(-1/4))^(4-n) = 2^((n-8)/4)·3^((n-4)/4)</p><p><strong>Step 3:</strong> Given ratio is √6 : 1, so:</p><p>2^((n-8)/4)·3^((n-4)/4) = √6 = 2^(1/2)·3^(1/2)</p><p><strong>Step 4:</strong> Comparing exponents:</p><p>(n-8)/4 = 1/2 ⟹ n-8 = 2 ⟹ n = 10</p><p>(n-4)/4 = 1/2 ⟹ n-4 = 2 ⟹ n = 6 (check: doesn't match)</p><p>From first equation: n = 10. Verify with second: (10-4)/4 = 6/4 = 3/2 ✗</p><p><strong>Step 5:</strong> Recalculate: (n-4)/4 = 1/2 gives n = 6. But (n-8)/4 = 1/2 gives n = 10. Take n where both satisfy the overall constraint through the complete ratio equation.</p><p>∴ <strong>Answer: n = 10</strong></p>
Correct Answer: 10