If α,β  be the roots of x2−x−1=0  and An=αn+βn , then AM of An−1  and An  is

The given equation is x2−x−1=0 Let α,β  are the roots given equation∴  Sum of the roots : α+β=1  and     Product of the roots : αβ=−1 Given that An=αn+βn AM of An−1  and An=An−1+An2                 =αn−1+βn−1+αn+βn2                 =αn−1(1+α)+βn−1(1+β)2                =αn−1(α2)+βn−1(β2)2          (∵α2=α+1   and     β2=β+1)                =12(αn+1+βn+1)                =12An+1