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Let Tₙ be the determinant | n-1, n-2, n-6; 2n-4, 2n-6, 2n-11; 3n-9, 3n-12, 3n-18 |, where n is a natural number. Which of the following statements are true? (A) Product(n=1 to n) Tₙ = 6ⁿ (B) Product(n=1 to 10) Tₙ = 60 (C) T_(n+1) / Tₙ = 2 (D) T₁₀₀ * T₁₀₁ = 48

1. (A) Product(n=1 to n) Tₙ = 6ⁿ
2. (B) Product(n=1 to 10) Tₙ = 60
3. (C) T_(n+1)/Tₙ = 2
4. (D) T₁₀₀ * T₁₀₁ = 48

Correct answer: (A) Product(n=1 to n) Tₙ = 6ⁿ

Solution

After subtracting 2*R1 from R2 and 3*R1 from R3, the determinant simplifies to a 3x3 with constant last two rows: R2 = (-2,-2,1) and R3 = (-6,-6,0); expanding gives Tₙ = 6 for all n. Hence the product of n terms = 6ⁿ (true), the product of 10 terms = 6¹⁰ (not 60), T_(n+1)/Tₙ = 1 (not 2), and T₁₀₀*T₁₀₁ = 36 (not 48).

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