Quadratic Equations
Recurrence from Quadratic Root Relation
nta_pyq_2023_apr
Grade 11
Question:
If $a$ and $b$ are roots of $x^2-7x-1=0$, then $\dfrac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}$ is equal to
Step-by-Step Solution
Key Concept: $a^2=7a+1$, so $a^4+1=51a^2$. Thus $a^{17}(a^4+1)=51a^{19}$ and similarly for $b$.
$\frac{51(a^{19}+b^{19})}{a^{19}+b^{19}}=51$.
Correct Answer: 51