<p>Let \((x + 10)^{50} + (x - 10)^{50} = a_0 + a_1x + a_2x^2 + \cdots + a_{50}x^{50}\), for all \(x \in \mathbb{R}\); then \(\dfrac{a_2}{a_0}\) is equal to ___________.</p>
Step-by-Step Solution
Key Concept: When expanding (x+10)^50 + (x-10)^50, only even powers of x survive due to symmetry (odd powers cancel). Use binomial theorem to find coefficients of x^0 and x^2 terms.
<p><strong>Step 1:</strong> Recognize symmetry. Since we have f(x) + f(-x) where f(x) = (x+10)^50, only even powers survive. This means a_1 = a_3 = a_5 = ... = 0.</p><p><strong>Step 2:</strong> Find a_0 using binomial theorem:</p><p>a_0 = coefficient of x^0 = 10^50 + (-10)^50 = 10^50 + 10^50 = 2·10^50</p><p><strong>Step 3:</strong> Find a_2 using binomial theorem:</p><p>(x+10)^50 = Σ C(50,k)x^k·10^(50-k)</p><p>(x-10)^50 = Σ C(50,k)x^k·(-10)^(50-k)</p><p>Coefficient of x^2 in (x+10)^50: C(50,2)·10^48</p><p>Coefficient of x^2 in (x-10)^50: C(50,2)·10^48</p><p>Therefore: a_2 = 2·C(50,2)·10^48 = 2·(50·49/2)·10^48 = 50·49·10^48</p><p><strong>Step 4:</strong> Calculate the ratio:</p><p>a_2/a_0 = (50·49·10^48)/(2·10^50) = (50·49)/(2·10^2) = 2450/200 = 12.5</p><p>∴ Answer: <strong>12.5</strong></p>
Correct Answer: 12.5