Quadratic Equations Question - None

Question

Given a and b are the roots of the equation $x^2 - 6x - 2 = 0$. Let $a_n = a^n - b^n$ for $n \geq 1$. Find $\frac{a_{10} - 2a_8}{2a_9}$.

Accepted Answer

Step 1: Since a and b are roots of $x^2 - 6x - 2 = 0$, we have:

$a^2 = 6a + 2$ and $b^2 = 6b + 2$

Step 2: Consider the expression:

$\frac{a_{10} - 2a_8}{2a_9} = \frac{a^{10} - b^{10} - 2(a^8 - b^8)}{2(a^9 - b^9)}$

Step 3: Factor the numerator:

$= \frac{a^8(a^2 - 2) - b^8(b^2 - 2)}{2(a^9 - b^9)}$

Step 4: Substitute $a^2 = 6a + 2$ and $b^2 = 6b + 2$:

$= \frac{a^8 \cdot 6a - b^8 \cdot 6b}{2(a^9 - b^9)}$

$= \frac{6(a^9 - b^9)}{2(a^9 - b^9)}$

$= \frac{6}{2} = 3$

∴ Answer is 3.

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