<p>If <span>\(\dfrac{1}{\sqrt{4x+1}}\left\{\left(\dfrac{1+\sqrt{4x+1}}{2}\right)^n - \left(\dfrac{1-\sqrt{4x+1}}{2}\right)^n\right\} = a_0 + a_1 x + \cdots + a_5 x^5\)</span>, then find the possible values of <span>\(n\)</span>.</p>
Step-by-Step Solution
Key Concept: Recognize that the expression simplifies to a polynomial of degree 5 in x when expanded using binomial theorem. The highest power of x appearing must come from the binomial expansions, and the coefficient of x^6 must vanish for the polynomial to have exactly 5 terms.
<p><strong>Step 1:</strong> Let u = √(4x+1). Then the expression becomes:</p><p>$$\frac{1}{u}\left[\left(\frac{1+u}{2}\right)^n - \left(\frac{1-u}{2}\right)^n\right]$$</p><p><strong>Step 2:</strong> Expand using binomial theorem. Note that (1+u)^n - (1-u)^n contains only odd powers of u (by symmetry), so:</p><p>$$\left(\frac{1+u}{2}\right)^n - \left(\frac{1-u}{2}\right)^n = \frac{1}{2^n}\sum_{k\text{ odd}} 2\binom{n}{k}u^k$$</p><p><strong>Step 3:</strong> Dividing by u = √(4x+1):</p><p>$$\frac{1}{u}\sum_{k\text{ odd}} \frac{2\binom{n}{k}}{2^n}u^k = \sum_{k\text{ odd}} \frac{2\binom{n}{k}}{2^n}u^{k-1}$$</p><p><strong>Step 4:</strong> Since u^m = (4x+1)^{m/2}, when we expand in powers of x, u^{k-1} contributes terms up to x^{(k-1)/2}. For the RHS to be a polynomial in x of degree ≤ 5, we need the highest odd k value to satisfy (k-1)/2 ≤ 5, so k-1 ≤ 10, giving k ≤ 11.</p><p><strong>Step 5:</strong> The coefficient of x^6 in the full expansion (from all terms) must equal zero. The term with u^11 gives contributions up to x^5, and u^13 gives contributions starting from x^6. For no x^6 term: either n < 13 or the x^6 coefficient from u^{13} must vanish.</p><p><strong>Step 6:</strong> When n = 11: highest odd power is 11, so u^11 gives x^5 (the last term). When n = 12: highest odd power is 11, still giving x^5 as maximum. When n = 13: u^13 appears and creates x^6 terms. When n = 11 or 12, the polynomial has exactly degree 5 with a₅ ≠ 0.</p><p>∴ Answer: <strong>n = 11 or n = 12</strong></p>
Correct Answer: n = 11 or 12