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junior college 1 | H2 Maths
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Date Posted: 3 years ago
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i) Since uₙ is a cubic polynomial in n, let uₙ = An³ + Bn² + Cn + D, where A,B,C,D are constants.
u₁ = A(1³) + B(1²) + C(1) + D → 32.1 = A + B + C + D
u₂ = A(2³) + B(2²) + C(2) + D→ 17 = 8A + 4B + 2C + D
u₃ = A(3³) + B(3²) + C(3) + D→ 0.7 = 27A + 9B + 3C + D
u₄ = A(4³) + B(4²) + C(4) + D→ -7.8 = 64A + 16B + 4C + D
Solving for these four equations (using GC or manually), A = 1.5, B = -9.6, C = 3.2, D = 37
∴ uₙ = 1.5n³ - 9.6n² + 3.2n + 37
ii) For uₙ > 555,
1.5n³ - 9.6n² + 3.2n + 37 > 555
Iteration method :
When n = 8 , u₈ = 216.2
When n = 9, u₉ = 381.7
When n = 10, u₁₀ = 609
When n = 11, u₁₁ = 907.1
When n = 12, u₁₂ = 1285
Minimum value of n = 10
u₁ = A(1³) + B(1²) + C(1) + D → 32.1 = A + B + C + D
u₂ = A(2³) + B(2²) + C(2) + D→ 17 = 8A + 4B + 2C + D
u₃ = A(3³) + B(3²) + C(3) + D→ 0.7 = 27A + 9B + 3C + D
u₄ = A(4³) + B(4²) + C(4) + D→ -7.8 = 64A + 16B + 4C + D
Solving for these four equations (using GC or manually), A = 1.5, B = -9.6, C = 3.2, D = 37
∴ uₙ = 1.5n³ - 9.6n² + 3.2n + 37
ii) For uₙ > 555,
1.5n³ - 9.6n² + 3.2n + 37 > 555
Iteration method :
When n = 8 , u₈ = 216.2
When n = 9, u₉ = 381.7
When n = 10, u₁₀ = 609
When n = 11, u₁₁ = 907.1
When n = 12, u₁₂ = 1285
Minimum value of n = 10
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