If λ,β≠$$0$$,and $$fn=$$λ$$n+$$βn and $$31+f11+f21+f11+f21+f31+f21+f31+f4=K1$$−α$$21$$−β$$2$$α−β$$2$$ Then $$K=$$

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If λ,β≠$$0$$,and $$fn=$$λ$$n+$$βn and $$31+f11+f21+f11+f21+f31+f21+f31+f4=K1$$−α$$21$$−β$$2$$α−β$$2$$ Then $$K=$$

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We $$have31+f(1)1+f(2)1+f(1)1+f(2)1+f(3)1+f(2)1+f(3)1+f(4)$$ ⇒$$31+$$α$$+$$β$$1+$$α$$2+$$β$$21+$$α$$+$$β$$1+$$α$$2+$$β$$21+$$α$$3+$$β$$31+$$α$$2+$$β$$21+$$α$$3+$$β$$31+$$α$$4+$$β$$4=11121$$αβ$$1$$α$$2$$β$$211121$$αα$$21$$ββ$$2$$                                                                    $$=11121$$αβ$$1$$α$$2$$β$$211121$$αβ$$1$$α$$2$$β$$2$$                                                                    $$=1-$$α$$2$$α$$-$$β$$2$$β$$-12$$                                                                    $$=k1-$$α$$21-$$β$$2$$α$$-$$β$$2$$ ⇒$$k=1$$

If $λ, β \neq 0,$ and $f (n) = λ^{n} + β^{n}$ and $|\begin{matrix} 3 & 1 + f (1) & 1 + f (2) \\ 1 + f (1) & 1 + f (2) & 1 + f (3) \\ 1 + f (2) & 1 + f (3) & 1 + f (4) \end{matrix}|$ $= K {(1 - α)}^{2} {(1 - β)}^{2} {(α - β)}^{2}$ Then K=

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If λ,β≠0,and fn=λn+βn and 31+f11+f21+f11+f21+f31+f21+f31+f4=K1−α21−β2α−β2 Then K=

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If $λ, β \neq 0,$ and $f (n) = λ^{n} + β^{n}$ and $|\begin{matrix} 3 & 1 + f (1) & 1 + f (2) \\ 1 + f (1) & 1 + f (2) & 1 + f (3) \\ 1 + f (2) & 1 + f (3) & 1 + f (4) \end{matrix}|$ $= K {(1 - α)}^{2} {(1 - β)}^{2} {(α - β)}^{2}$ Then K=