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If a(n+1)=1/(1-an) for n>=1 and a3=a1...

If `a_(n+1)=1/(1-a_n)` for `n>=1` and `a_3=a_1`. then find the value of `(a_2001)^2001`.

Text Solution

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`a_2=1/(1-a_1)`
`a_3=1/(1-a_2)=(1-a_1)/(-a_1)`
`-a_1^2=(1-a_1)`
`a_1^2-a_1+1=0`
`a_1=(1pmsqrt(1-4))/2=e^(ipi/3),e^(i-pi/3`
`a_1=a_3=a_5=a_7...=a_2001=e^(ipi/3`
`(a_2001)^2001=(e^ipmpi/3)^2001`
`=e^(pmi(667pi)`
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