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The Lucas sequence Lₙ is defined by the recurrence relation: Lₙ = Lₙ₋₁ + Lₙ₋₂, for n ≥ 3, with L₁ = 1 and L₂ = 3. Which one of the options given is TRUE?

1. Lₙ = ((1 + √5)/2)ⁿ + ((1 - √5)/2)ⁿ
2. Lₙ = ((1 + √5)/2)ⁿ - ((1 - √5)/3)ⁿ
3. Lₙ = ((1 + √5)/2)ⁿ + ((1 - √5)/3)ⁿ
4. Lₙ = ((1 + √5)/2)ⁿ - ((1 - √5)/2)ⁿ

Correct answer: Lₙ = ((1 + √5)/2)ⁿ + ((1 - √5)/2)ⁿ

Solution

For L1=1, L2=3 with Ln=Ln-1+Ln-2, the closed form is Ln = ((1+sqrt5)/2)^n + ((1-sqrt5)/2)^n, which gives L1=1 and L2=3. That is index 0, not the stored difference form (index 3) which would give a sqrt5 factor.

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