Quadratic Equations
Recurrence Relation from Quadratic Roots
nta_pyq_2024_jan
Grade 11
Question:
If $\alpha,\beta$ are the roots of the equation $x^2-x-1=0$ and $S_n=2023\alpha^n+2024\beta^n$, then
$2S_{12}=S_{11}+S_{10}$
$S_{12}=S_{11}+S_{10}$
$2S_{11}=S_{12}+S_{10}$
$S_{11}=S_{10}+S_{12}$
Step-by-Step Solution
Key Concept: $\alpha^2=\alpha+1$ and $\beta^2=\beta+1$. So $\alpha^n=\alpha^{n-1}+\alpha^{n-2}$ and $\beta^n=\beta^{n-1}+\beta^{n-2}$. Thus $S_n=S_{n-1}+S_{n-2}$.
$S_{12}=S_{11}+S_{10}$.
Correct Answer: 2