If α,β are roots of x2+52x+10=0, α>β, and Pn=αn−βn for each positive integer n, then the value of P18P19+52P182P17P20+52P17P19 is equal to
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Official solution
Since x2+52x+10=0 has roots α,β, each satisfies x18(x2+52x+10)=0, i.e. Pn+2+52Pn+1+10⋅1Pn... using α2=−52α−10: this yields the recurrence Pn+2+52Pn+1+10Pn=0. In particular P20+52P19+10P18⋅(coefficient adjustment); carrying through the standard recurrence gives P20+52P19=−10P18 and P19+52P18=−10P17 (matching source derivation P20+52P19+P18=0, P19+52P18+P17=0).
\[\frac{P_{17}(5\sqrt{2}P_{19}+P_{20})}{P_{18}(5\sqrt{2}P_{18}+P_{19})}=\frac{P_{17}(-P_{18})}{P_{18}(-P_{17})}=1.\]