Question

If $\alpha, \beta$ are the roots of the equation $x^2 - x - 1 = 0$ and $S_n = 2023 \alpha^n + 2024 \beta^n$, then:

• A. $S_{12} = S_1 + S_{10}$
• B. $2S_{11} = S_{12} + S_{10}$
• C. $S_{11} = S_{10} + S_{12}$
• D. $S_1 + S_{10} = S_{12}$

Answer 1

Answer: (B)

$2S_{11} = S_{12} + S_{10}$

Answer 2

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

The relation between $\alpha$ and $\beta$ is:
$\alpha^2 = \alpha + 1, \quad \beta^2 = \beta + 1.$

The sequence $S_n$ is defined as:
$S_n = 2023\alpha^n + 2024\beta^n.$

Using the recurrence relation for the roots:
$S_{n+2} = S_{n+1} + S_n.$

Applying this for $n = 10$:
$S_{12} = S_{11} + S_{10}.$

The Correct answer is: $2S_{11} = S_{12} + S_{10}$