Question

If \( \alpha, \beta \) are the roots of \( x^2 - 10x - 8 = 0 \), with \( \alpha>\beta \), and \( a_n = \alpha^n - \beta^n \), then the value of: \[ \frac{a_{10} - 8a_8}{5a_9} \] is:

• A. -3
• B. 3
• C. -2
• D. 2

Hint:

When dealing with sequences defined by powers of roots, use the recurrence relation derived from the original quadratic to simplify higher terms.

Answer 1

The Correct Option is D

Given recurrence relation: \[ a_n = \alpha^n - \beta^n \] Roots of \( x^2 - 10x - 8 = 0 \) are: \[ \alpha = 5 + \sqrt{33},\quad \beta = 5 - \sqrt{33} \] We use identity for such sequences: \[ a_{n+1} = 10a_n + 8a_{n-1} \] Apply this recursively to find a pattern: We are given: \[ \frac{a_{10} - 8a_8}{5a_9} \] Let’s denote recurrence: \[ a_{10} = 10a_9 + 8a_8 \Rightarrow a_{10} - 8a_8 = 10a_9 \Rightarrow \frac{a_{10} - 8a_8}{5a_9} = \frac{10a_9}{5a_9} = 2 \]