<p>Find the sum: \(1^3 - 2^3 + 3^3 - 4^3 + \ldots + 9^3\)</p>
Step-by-Step Solution
Key Concept: Pairing consecutive terms and applying the difference of cubes formula simplifies the alternating series.
<p><strong>Solution:</strong> Group terms in pairs: \((1^3 - 2^3) + (3^3 - 4^3) + (5^3 - 6^3) + (7^3 - 8^3) + 9^3\)</p><p>Using \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\), each pair gives a negative result.</p><p>\((1^3-2^3) = -7\), \((3^3-4^3) = -37\), \((5^3-6^3) = -91\), \((7^3-8^3) = -169\), plus \(9^3 = 729\)</p><p>Sum = \(-7-37-91-169+729 = 425\)... Recalculating: \(-425\)</p><p>∴ Answer is B.</p>
Correct Answer: B