1) Let p,q be integers and let α ,β be the roots of the equation $x^{2}-x-1=0$ where α ≠β, For n=0,1,2...... Let $a_{n}=p\alpha^{n}+q\beta^{n}$ ( If a and b are rational numbers and $a+b\sqrt{5}=0$, then a=0=b) a12= A) $a_{11}+2a_{10}$ B) $2a_{11}+a_{10}$ C) $a_{11}-a_{10}$ D) $a_{11}+a_{10}$ Answer: Option DExplanation:$\alpha^{2}=\alpha+1$ $\beta^{2}=\beta+1$ $a_{n}=p\alpha^{n}+q\beta^{n}$ $=p(\alpha^{n-1}+\alpha^{n-2})+q(\beta^{n-1}+\beta^{n-2})$ $=a_{n-1}+a_{n-2}$ $\therefore$ $a_{12}=a_{11}+a_{10}$
