Let p, q be integers and let  α,β be the roots of x2−x−1=0,  where  α≠β.  For  n=0,1,2,… Let  an=pαn+qβ2Then,

It is given that  α and β are roots of x2−x−1=0∴ α+β=1  and  αβ=−1 We have an=pαn+qβn∴     an+1=pαn+1+qβn+1⇒     an+1=pαn+qβn(α+β)−αβpαn−1+qβn−1⇒     an+1=an+an−1                                 [∵α+β=1,αβ=−1]