Discussion Forum

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=

Answer:

Explanation:

$\alpha^{2}=\alpha+1$

$\beta^{2}=\beta+1$

$a_{n}=p\alpha^{n}+q\beta^{n}$

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

$=a_{n-1}+a_{n-2}$

$\therefore$   $a_{12}=a_{11}+a_{10}$