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p(x) is a function for which p(1)=1, p(2)=2, p(3)=3 and p(x+1)= p(x-2)p(x-1)+1/p(x) for which x greater than or equal to 3
Find p(6)

Respuesta :

LammettHash

Use the recursive definition of [tex]p(x)[/tex]: for [tex]x\ge3[/tex],

[tex]p(x+1)=p(x-2)p(x-1)+\dfrac1{p(x)}[/tex]

So we have

[tex]x=3\implies p(4)=p(1)p(2)+\dfrac1{p(3)}=2+\dfrac13=\dfrac73[/tex]

[tex]x=4\implies p(5)=p(2)p(3)+\dfrac1{p(4)}=6+\dfrac37=\dfrac{45}7[/tex]

[tex]x=5\implies p(6)=p(3)p(4)+\dfrac1{p(5)}=7+\dfrac7{45}=\boxed{\dfrac{322}{45}}[/tex]

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