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

To solve this problem, we first need to understand the properties of the roots \( \alpha \) and \( \beta \) of the given quadratic equation \( x^2 - x - 1 = 0 \).

The solutions to this equation can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -1 \), and \( c = -1 \). 

This gives us:

\(x = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}\)

Thus, the roots are: \( \alpha = \frac{1 + \sqrt{5}}{2} \) and \( \beta = \frac{1 - \sqrt{5}}{2} \).

Next, we use the relationships for powers of roots of quadratic equations. Since \( \alpha \) and \( \beta \) are roots of the equation \( x^2 - x - 1 = 0 \), they satisfy:

• \(\alpha^2 = \alpha + 1\)
• \(\beta^2 = \beta + 1\)

These recursive relationships can be used to express higher powers as follows:

• \(\alpha^n = \alpha^{n-1} + \alpha^{n-2}\)
• \(\beta^n = \beta^{n-1} + \beta^{n-2}\)

The sequence relations for \( S_n = 2023 \alpha^n + 2024 \beta^n \) can be used to express each term in terms of previous terms:

• \(S_n = 2023 \alpha^n + 2024 \beta^n = 2023(\alpha^{n-1} + \alpha^{n-2}) + 2024(\beta^{n-1} + \beta^{n-2})\)

This can be expanded to find:

• \(S_n = 2023 \alpha^{n-1} + 2024 \beta^{n-1} + 2023 \alpha^{n-2} + 2024 \beta^{n-2}\)
• \(S_n = S_{n-1} + S_{n-2}\)

Using this relation, let's analyze the options:

• Option 1: \( S_{12} = S_1 + S_{10} \) - This implies adding a non-consecutive term, which doesn't match the recursive definition.
• Option 2: \( 2S_{11} = S_{12} + S_{10} \) - This option implies a direct relationship with a pattern that fits the recursive formula expressed above.
• Option 3: \( S_{11} = S_{10} + S_{12} \) - This implies a reverse recursion which isn't supported by our recursive expression.
• Option 4: \( S_1 + S_{10} = S_{12} \) - Similar to Option 1, this does not match the pattern.

Thus, the correct answer is Option 2: \(2S_{11} = S_{12} + S_{10}\), as this expression is consistent with the recursive relationships derived for \( S_n \).

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