Multiple choice

Let $\alpha$ and $\beta$ be the roots of equation $x^{2}-6x-2=0$. If $a_{m}=\alpha^{n}-\beta^{n}$, for $n \ge 1$, then the value of $\dfrac {a_{10}-2a_{8}}{2a_{9}}$ is equal to:

  1. $3$
  2. $-3$
  3. $6$
  4. $-6$
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A Correct answer
Explanation

Given x^2 - 6x - 2 = 0, we have alpha^2 = 6alpha + 2 and beta^2 = 6beta + 2. Multiplying by alpha^(n-2) and beta^(n-2) respectively leads to the recurrence relation a_n = 6a_(n-1) + 2a_(n-2). Rearranging gives (a_n - 2a_(n-2)) / a_(n-1) = 6. For n=10, (a_10 - 2a_8) / a_9 = 6. Dividing by 2 gives 3.

AI explanation

From the given equation, the sum of roots alpha + beta = 6 and the product of roots alpha*beta = -2. Rearranging gives alpha^2 = 6*alpha + 2 and beta^2 = 6*beta + 2, which can be generalized as a_n = 6*a_(n-1) + 2*a_(n-2). Substituting this recurrence relation into the numerator yields a_10 - 2*a_8 = (6*a_9 + 2*a_8) - 2*a_8 = 6*a_9. Dividing by the denominator 2*a_9 results in 6*a_9 / 2*a_9 = 3.

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