<p>Let <i>p</i> be the coefficient of <i>x</i><sup>2</sup> in the expansion <i>(1 + x)(1 – 3x)(1 + 5x)(1 – 7x)...(1 – 23x)(1 + 25x)</i>, then the sum of the digits of |<i>p</i>| is equal to</p>
Step-by-Step Solution
Key Concept: The coefficient of x² in a product of linear terms (1 + aₙx) comes from selecting x from exactly two factors and 1 from the rest. We need to identify all pairs of factors and sum the products of their coefficients.
<p><strong>Step 1:</strong> Identify the structure of the product. We have factors (1 + x)(1 - 3x)(1 + 5x)(1 - 7x)...(1 - 23x)(1 + 25x). The coefficients of x are: 1, -3, 5, -7, 9, -11, 13, -15, 17, -19, 21, -23, 25.</p><p><strong>Step 2:</strong> Count the factors. The pattern shows odd numbers: 1, 3, 5, 7, ..., 23, 25. This is 13 factors total (from (2k-1) where k = 1 to 13).</p><p><strong>Step 3:</strong> The coefficient of x² comes from choosing exactly two factors and taking x from each, with 1 from all others. If we select factors (1 + aᵢx) and (1 + aⱼx) where i < j, the contribution to x² is aᵢ·aⱼ.</p><p><strong>Step 4:</strong> The coefficient p is the sum of all products aᵢ·aⱼ for i < j, which equals:</p><p><strong>p = Σᵢ<ⱼ aᵢaⱼ</strong></p><p>Using the identity: (Σaᵢ)² = Σaᵢ² + 2Σᵢ<ⱼ aᵢaⱼ</p><p>Therefore: Σᵢ<ⱼ aᵢaⱼ = [(Σaᵢ)² - Σaᵢ²]/2</p><p><strong>Step 5:</strong> Calculate Σaᵢ = 1 - 3 + 5 - 7 + 9 - 11 + 13 - 15 + 17 - 19 + 21 - 23 + 25</p><p>Pairing consecutive terms: (1 - 3) + (5 - 7) + (9 - 11) + (13 - 15) + (17 - 19) + (21 - 23) + 25</p><p>= -2 - 2 - 2 - 2 - 2 - 2 + 25 = -12 + 25 = 13</p><p><strong>Step 6:</strong> Calculate Σaᵢ² = 1² + 3² + 5² + 7² + 9² + 11² + 13² + 15² + 17² + 19² + 21² + 23² + 25²</p><p>= 1 + 9 + 25 + 49 + 81 + 121 + 169 + 225 + 289 + 361 + 441 + 529 + 625 = 3325</p><p><strong>Step 7:</strong> Calculate (Σaᵢ)² = 13² = 169</p><p><strong>Step 8:</strong> Calculate p = [169 - 3325]/2 = -3156/2 = -1578</p><p><strong>Step 9:</strong> Find |p| = 1578. The sum of digits = 1 + 5 + 7 + 8 = 21</p><p><strong>∴ Answer: 21</strong></p>
Correct Answer: 21