If α, β are the roots of the equation x2 - x - 1 = 0 and Sn = 2023 αn + 2024 βn, then:
-
S11 = S10 + S12
-
2S11 = S12 + S10
-
2S12 = S11 + S10
-
S12 = S11 + S10
Since alpha and beta are roots of x^2 - x - 1 = 0, they satisfy x^2 = x + 1. Multiplying by x^(n-2) gives x^n = x^(n-1) + x^(n-2). This linear recurrence holds for both alpha and beta, so S_n = 2023*alpha^n + 2024*beta^n also satisfies S_n = S_(n-1) + S_(n-2).
Because alpha and beta are roots of x^2 - x - 1 = 0, each satisfies the relation r^n = r^(n-1) + r^(n-2). Multiply this relation by 2023 for alpha and by 2024 for beta, then add the two resulting equations. This yields 2023*alpha^n + 2024*beta^n = (2023*alpha^(n-1) + 2024*beta^(n-1)) + (2023*alpha^(n-2) + 2024*beta^(n-2)), which simplifies to S_n = S_(n-1) + S_(n-2). Therefore, S_12 = S_11 + S_10.