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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. −1
3. αβ
4. 1/(αβ)

Correct answer: αβ

Solution

The determinant of the matrix is structured such that it incorporates the terms from the function f(n), which is defined as α^n + β^n. The factor K is derived from the properties of the roots α and β, leading to the conclusion that K equals the product of the roots, αβ, as it reflects the scaling of the determinant based on the roots of the polynomial.

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