Sequences & Series
Power sum from roots of quadratic
MMTS_Full_Test_17
Grade 12
Question:
Let $\alpha,\beta$ be roots of $x^2-10x+2=0$. Value of $\dfrac{\alpha^{2028}+\beta^{2028}+8\alpha^{2022}+8\beta^{2022}}{\alpha^{2025}+\beta^{2025}}$ is
Step-by-Step Solution
Key Concept: Use recurrence $P_n=\alpha^n+\beta^n$: $P_n=10P_{n-1}-2P_{n-2}$. Expression $=\dfrac{P_{2028}+8P_{2022}}{P_{2025}}$.
Value $=940$.
Correct Answer: 940