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If α, β ≠ 0, and f(n) = α^n + β^n and | 3 1+f(1) 1+f(2) | | 1+f(1) 1+f(2) 1+f(3) | | 1+f(2) 1+f(3) 1+f(4) | = K(1−α)^2(1−β)^2(α−β)^2, then K is equal to

1. −1
2. αβ
3. 1/(αβ)
4. 1

Correct answer: 1

Solution

The determinant of the given matrix simplifies to a form that reveals K as a constant factor, specifically 1, when considering the structure of the functions involved and their relationships. This is consistent with the properties of determinants and the specific forms of f(n) in the context of the problem.

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