<p>Given \((x+10)^{50} + (x-10)^{50} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{50}x^{50}\), find the value of \(\dfrac{a_2}{a_0}\).</p>
Step-by-Step Solution
Key Concept: When expanding (x+10)^50 + (x-10)^50, all odd-power terms cancel due to symmetry, leaving only even-power coefficients. Extract a₀ and a₂ by evaluating at specific values of x or using the binomial theorem directly for even terms.
<p><strong>Step 1:</strong> Expand using binomial theorem. Only even powers survive in the sum:</p><p>(x+10)⁵⁰ + (x-10)⁵⁰ = 2[C(50,0)x⁰·10⁵⁰ + C(50,2)x²·10⁴⁸ + C(50,4)x⁴·10⁴⁶ + ...]</p><p><strong>Step 2:</strong> Identify coefficients:</p><p>a₀ = 2·C(50,0)·10⁵⁰ = 2·10⁵⁰</p><p>a₂ = 2·C(50,2)·10⁴⁸ = 2·[50·49/2]·10⁴⁸ = 2·(1225)·10⁴⁸ = 2450·10⁴⁸</p><p><strong>Step 3:</strong> Calculate the ratio:</p><p>a₂/a₀ = (2450·10⁴⁸)/(2·10⁵⁰) = 2450/(2·100) = 2450/200 = 12.25</p><p>∴ Answer: <strong>12.25</strong></p>
Correct Answer: 12.25