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

The Correct Option is B

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: \( 2S_{11} = S_{12} + S_{10} \)