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Question Number 87648 by john santu last updated on 05/Apr/20

$$\mathrm{the}\mathrm{sequence}{\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,...\mathrm{satisfies}$$ $$\mathrm{the}\mathrm{relation}{\mathrm{a}}_{\mathrm{n}+1}={\mathrm{a}}_{\mathrm{n}}+{\mathrm{a}}_{\mathrm{n}-1},\mathrm{for}$$ $$\mathrm{n}>1.\mathrm{given}\mathrm{that}{\mathrm{a}}_{20}=6765\mathrm{and}$$ $${\mathrm{a}}_{18}=2584\mathrm{what}\mathrm{is}{\mathrm{a}}_{16}$$

Commented byjohn santu last updated on 05/Apr/20

$${\mathrm{a}}_{20}={\mathrm{a}}_{19}+{\mathrm{a}}_{18}$$ $$={\mathrm{a}}_{18}+{\mathrm{a}}_{17}+{\mathrm{a}}_{18}$$ $$=3\times {\mathrm{a}}_{18}-{\mathrm{a}}_{16}$$ $$\Rightarrow {\mathrm{a}}_{16}=3\times {\mathrm{a}}_{18}-{\mathrm{a}}_{20}$$ $$\Rightarrow {\mathrm{a}}_{16}=3\times 2584-6765=987$$

Commented byjohn santu last updated on 05/Apr/20

$$\mathrm{so}\mathrm{for}\mathrm{this}\mathrm{question}$$ $${\mathrm{x}}^2-\mathrm{x}-1=0$$ $$\mathrm{x}=\frac{1\pm \sqrt{5}}{2}$$ $${\mathrm{a}}_{\mathrm{n}}=\mathrm{A}{\left(\frac{1+\sqrt{5}}{2}\right)}^{\mathrm{n}}+\mathrm{B}{\left(\frac{1-\sqrt{5}}{2}\right)}^{\mathrm{n}}$$

Commented bymr W last updated on 05/Apr/20

$$yes$$

Commented byjohn santu last updated on 05/Apr/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

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