Quadratic Equation and Inequalities Part 1

Given that, α+β=1,αβ=−1 Let α,β be roots of quadratic equation, then the quadratic equation be x2−x−1=0
‌ Now, ‌‌α2−α−1=0
⇒‌α2=α+1 . . . (i)
Similarly, β2=β+1 . . . (ii)
Multiply αn−1 in Eq. (i), we get
αn+1=αn+αn−1 . . . (iii)
Multiply βn−1 in Eq. (ii), we get
βn+1=βn+βn−1 . . . (iv)
Add Eqs. (iii) and (iv), we get
αn+1+βn+1‌‌=(αn+βn)+(αn−1+βn−1)
pn+1‌‌=pn+pn−1
29‌‌=pn+11⇒Pn=18
pn2‌‌=(18)2=324