Question

Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation, $x^2-x-1=0$, where $\alpha \ne \beta$. For $n=0,1,2,...$ let $a_n=p{\alpha }^n+q{\beta }^n$.
FACT : If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0$, then $a=0=b$.
$a_{12}=$

• A. $a_{11}-a_{10}$
• B. $a_{11}+a_{10}$
• C. $2a_{11}+a_{10}$
• D. $a_{11}+2a_{10}$

Answer 1

The correct option is B $a_{11}+a_{10}$

$a_{n+2}-a_{n+1}-a_n=p{\alpha }^n({\alpha }^2-\alpha -1)+q{\beta }^n({\beta }^2-\beta -1)=0$

So,

$a_{12}-a_{11}-a_{10}=0$

$a_{12}=a_{11}+a_{10}$