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Question Number 135797 by benjo_mathlover last updated on 16/Mar/21

$$$$How do you solve f(3) when f(-1) = -1 and f(0) =1 and f(x) =f(x-1)-2f(x-2)?

Answered by liberty last updated on 17/Mar/21

$$f\left(x\right)=f(x-1)-2f(x-2)$$$$replacex-2byxweget$$$$2f\left(x\right)=f(x+1)-f(x+2)$$$$$$$$\left(1\right)x=-1\Rightarrow 2f(-1)=f\left(0\right)-f\left(1\right)$$$$\Rightarrow -2=1-f\left(1\right);f\left(1\right)=3$$$$$$$$\left(2\right)x=0\Rightarrow 2f\left(0\right)=f\left(1\right)-f\left(2\right)$$$$\Rightarrow 2=3-f\left(2\right);f\left(2\right)=1$$$$$$$$\left(3\right)x=1\Rightarrow 2f\left(1\right)=f\left(2\right)-f\left(3\right)$$$$\Rightarrow 6=1-f\left(3\right);f\left(3\right)=-5$$

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