Let αand βbe two real numbers such thatα+β=1 and.αβ=−1. Let pn=αn+βn,Pn−1=11and Pn+1=29 for some integer n≥1. Then, the value of pn2 is _____

Given α,β are the roots of the equation x2-x-1=0 because α+β=1 and αβ=-1 α2−α−1=0⇒α2=α+1⇒αn+1=αn+αn−1 and  β2−β−1=0⇒β2=β+1⇒βn+1=βn+βn−1Hence, αn+1+βn+1=αn+βn+1+αn−1+βn−1Pn+1=Pn+Pn−129=Pn+11Pn=18Therefore, Pn2=324