If α,β are the roots of x2-ax+b=0 and if αn+βn=Vn, then

Multiplying x2-ax+b=0 by xn-1xn+1-axn+bxn-1=0    ...(i)α,β are roots of x2-ax+b=0, therefore they will satisfy (i) also αn+1-aαn+bαn-1=0    ...(ii)and βn+1-αβn+bβn-1=0     ...(iii)adding (ii) and (iii)(αn+1+βn+1)-a(αn+βn)+b(αn-1+βn-1)=0or Vn+1=aVn-bVn-1(Given αn+βn=Vn)