Question-87767

Question Number 87767 by shshdhdhdhh@gmail.com last updated on 06/Apr/20

Commented by MJS last updated on 06/Apr/20

there′s not a single correct answer. it′s possible to find at least one function satisfying a chosen value in the empty field and all other fields. an easy example: find the next number: 1, 1, 2, 3, 5, 8, ? let ?=α then we can find a polynome f(x)=c_6 x^6 +c_5 x^5 +c_4 x^4 +c_3 x^3 +c_2 x^2 +c_1 x+c_0 with f(1)=1, f(2)=1, f(3)=2, f(4)=3, f(5)=5, f(6)=8 and f(7)=α ∀α∈C

$$\mathrm{there}'\mathrm{s}\:\mathrm{not}\:\mathrm{a}\:\mathrm{single}\:\mathrm{correct}\:\mathrm{answer}.\:\mathrm{it}'\mathrm{s} \\ $$$$\mathrm{possible}\:\mathrm{to}\:\mathrm{find}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{function}\:\mathrm{satisfying} \\ $$$$\mathrm{a}\:\mathrm{chosen}\:\mathrm{value}\:\mathrm{in}\:\mathrm{the}\:\mathrm{empty}\:\mathrm{field}\:{and}\:\mathrm{all}\:\mathrm{other} \\ $$$$\mathrm{fields}. \\ $$$$\mathrm{an}\:\mathrm{easy}\:\mathrm{example}:\:\mathrm{find}\:\mathrm{the}\:\mathrm{next}\:\mathrm{number}: \\ $$$$\mathrm{1},\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{5},\:\mathrm{8},\:? \\ $$$$\mathrm{let}\:?=\alpha \\ $$$$\mathrm{then}\:\mathrm{we}\:\mathrm{can}\:\mathrm{find}\:\mathrm{a}\:\mathrm{polynome} \\ $$$${f}\left({x}\right)={c}_{\mathrm{6}} {x}^{\mathrm{6}} +{c}_{\mathrm{5}} {x}^{\mathrm{5}} +{c}_{\mathrm{4}} {x}^{\mathrm{4}} +{c}_{\mathrm{3}} {x}^{\mathrm{3}} +{c}_{\mathrm{2}} {x}^{\mathrm{2}} +{c}_{\mathrm{1}} {x}+{c}_{\mathrm{0}} \\ $$$$\mathrm{with}\:{f}\left(\mathrm{1}\right)=\mathrm{1},\:{f}\left(\mathrm{2}\right)=\mathrm{1},\:{f}\left(\mathrm{3}\right)=\mathrm{2},\:{f}\left(\mathrm{4}\right)=\mathrm{3}, \\ $$$${f}\left(\mathrm{5}\right)=\mathrm{5},\:{f}\left(\mathrm{6}\right)=\mathrm{8}\:\mathrm{and}\:{f}\left(\mathrm{7}\right)=\alpha \\ $$$$\forall\alpha\in\mathbb{C} \\ $$

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