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 address this problem, we begin by examining the properties of the roots \( \alpha \) and \( \beta \) of the quadratic equation \( x^2 - x - 1 = 0 \).

The roots of this equation are determined using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), with \( a = 1 \), \( b = -1 \), and \( c = -1 \).

The calculation yields:

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

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

Next, we apply the established relationships for powers of roots of quadratic equations. As \( \alpha \) and \( \beta \) are roots of \( x^2 - x - 1 = 0 \), they satisfy the following identities:

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

These recursive relationships enable the expression of higher powers of the roots:

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

For the sequence defined as \( S_n = 2023 \alpha^n + 2024 \beta^n \), each term can be expressed in relation to preceding terms:

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

Expanding this expression leads to:

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

Utilizing this derived recurrence relation, we evaluate the given options:

• Option 1: \( S_{12} = S_1 + S_{10} \) - This formulation involves adding non-consecutive terms, deviating from the established recursive definition.
• Option 2: \( 2S_{11} = S_{12} + S_{10} \) - This option presents a relationship that aligns with the recursive formula previously derived.
• Option 3: \( S_{11} = S_{10} + S_{12} \) - This suggests a reversed recursion, which is not supported by our derived recursive expression.
• Option 4: \( S_1 + S_{10} = S_{12} \) - Analogous to Option 1, this does not conform to the identified pattern.

Therefore, the accurate selection is Option 2: \(2S_{11} = S_{12} + S_{10}\), as this equation is consistent with the recursive relationships established for \( S_n \).