Quadratic Equations
Recurrence from Roots — Linear Combination of Power Sums
nta_pyq_2024_apr
Grade 11
Question:
Let $\alpha,\beta$; $\alpha>\beta$, be the roots of the equation $x^2-\sqrt{2}x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n$, $n\in\mathbb{N}$. Then $(11\sqrt{3}-10\sqrt{2})P_{10}+(11\sqrt{2}+10)P_{11}-11P_{12}$ is equal to
$10\sqrt{3}P_9$
$11\sqrt{3}P_9$
$10\sqrt{2}P_9$
$11\sqrt{2}P_9$
Step-by-Step Solution
Key Concept: From the recurrence $P_{n+2}-\sqrt{2}P_{n+1}-\sqrt{3}P_n=0$. Use $P_{12}=\sqrt{2}P_{11}+\sqrt{3}P_{10}$ and $P_{11}=\sqrt{2}P_{10}+\sqrt{3}P_9$.
Using recurrences at $n=10$ and $n=9$: expression $=10\sqrt{3}P_9$.
Correct Answer: 1