Quadratic Equations
Power Sum Recurrence — Ratio
nta_pyq_2024_apr
Grade 11
Question:
Let $\alpha,\beta$ be roots of $x^2+\sqrt{2}x-8=0$. If $U_n=\alpha^n+\beta^n$, then $\dfrac{U_{10}+\sqrt{2}U_9}{2U_8}$ is equal to
Step-by-Step Solution
Key Concept: Since $\alpha^2=-\sqrt{2}\alpha+8$ and $\beta^2=-\sqrt{2}\beta+8$: $\alpha^8(\alpha^2+\sqrt{2}\alpha)+\beta^8(\beta^2+\sqrt{2}\beta)=8\alpha^8+8\beta^8$.
$U_{10}+\sqrt{2}U_9=8U_8$. Ratio $=4$.
Correct Answer: 4