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Let a1 = 50 and let the sequence a1, a2, a3,... satisfy n(aₙ - aₙ₊₁) = n³ + n² - aₙ for all n in N. Which of the following statements are true?

1. a₁₆ / 16 = -70
2. a₁₁ = -55
3. a₁₀ = 50
4. a₁₂ / 12 = -17

Correct answer: a₁₆ / 16 = -70

Solution

Rewriting: aₙ(n+1) - n*aₙ₊₁ = n²(n+1). Dividing by n(n+1): aₙ/n - aₙ₊₁/(n+1) = n. Let bₙ = aₙ/n. Then bₙ - bₙ₊₁ = n, which telescopes: bₙ = b₁ - (1+2+...+(n-1)) = 50 - n(n-1)/2. So aₙ = n*bₙ = n(50 - n(n-1)/2). Checking: a₁₆/16 = 50 - 120 = -70 [A: TRUE]. a₁₁ = 11(50-55) = -55 [B: TRUE]. a₁₀ = 10(50-45) = 50 [C: TRUE]. a₁₂/12 = 50-66 = -16, not -17 [D: FALSE].

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