If α and β are the roots of the equation x² - 6x - 2 = 0, and if an = αⁿ - βⁿ for n ≥ 1, find the value of (a₁₀ - — JEE Mathematics
If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 6x - 2 = 0$, and if $a_n = \alpha^n - \beta^n$ for $n \ge 1$, find the value of $\frac{a_{10} - 2a_8}{2a_9}$.
1 Answer
Since $\alpha$ and $\beta$ are the roots of $x^2 - 6x - 2 = 0$, they satisfy the equation:
$$\alpha^2 - 6\alpha - 2 = 0 \implies \alpha^2 - 2 = 6\alpha$$
$$\beta^2 - 6\beta - 2 = 0 \implies \beta^2 - 2 = 6\beta$$
Multiply the first equation by $\alpha^8$ and the second equation by $\beta^8$:
$$\alpha^{10} - 2\alpha^8 = 6\alpha^9$$
$$\beta^{10} - 2\beta^8 = 6\beta^9$$
Subtracting the second equation from the first:
$$(\alpha^{10} - \beta^{10}) - 2(\alpha^8 - \beta^8) = 6(\alpha^9 - \beta^9)$$
By definition, $a_n = \alpha^n - \beta^n$, so:
$$a_{10} - 2a_8 = 6a_9$$
Dividing both sides by $2a_9$:
$$\frac{a_{10} - 2a_8}{2a_9} = \frac{6a_9}{2a_9} = 3$$
Answer: 3