If α, β are the roots of the equation, x ^ 2 - x - 1 = 0 and S _ n...

Given: $\alpha, \beta$ are roots of ${x}^{2} - x - 1 = 0$.

$⇒ {\alpha}^{2} - \alpha - 1 = 0$

$⇒ {\alpha}^{2} = \alpha + 1 . . . \left(i\right)</p><p>Also,${S}_{n} = 2023{\alpha}^{n} + 2024{\beta}^{n}$</p><p>$⇒ {S}_{n - 1} + {S}_{n - 2} = 2023{\alpha}^{n - 1} + 2024{\beta}^{n - 1} + 2023{\alpha}^{n - 2} + 2024{\beta}^{n - 2}$</p><p>$⇒ {S}_{n - 1} + {S}_{n - 2} = 2023{\alpha}^{n - 2}\left(1 + \alpha\right) + 2024{\beta}^{n - 2}\left(1 + \beta\right)

Using equation \left(i\right),

$⇒ {S}_{n - 1} + {S}_{n - 2} = 2023{\alpha}^{n - 2}\left({\alpha}^{2}\right) + 2024{\beta}^{n - 2}\left({\beta}^{2}\right)</p><p>$⇒ {S}_{n - 1} + {S}_{n - 2} = 2023{\alpha}^{n} + 2024{\beta}^{n}$</p><p>$⇒ {S}_{n - 1} + {S}_{n - 2} = {S}_{n}$</p><p>Putting,$n = 12$</p><p>$⇒ {S}_{12} = {S}_{11} + {S}_{10}$</p>$