If α, β ≠ 0, and f(n) = αⁿ + βⁿ 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:

Question

Accepted Answer

1. With D = [[1,1,1],[1,a,b],[1,a^2,b^2]], the given matrix equals D D^T, so its determinant = (det D)^2 = ((a-1)(b-1)(b-a))^2 = (1-a)^2(1-b)^2(a-b)^2. Hence K=1.

If α, β ≠ 0, and f(n) = αⁿ + βⁿ 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:

Solution

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