If `alpha` and `beta` are the roots of the equation `x^2-2x+4=0`, prov – InterviewSolution

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1.	If `alpha` and `beta` are the roots of the equation `x^2-2x+4=0`, prove that `alpha^n+beta^n=2^(n+1)cos(npi/3)`
Answer» `x^2-2x+4=0` `(x-1)^2+3=0` `(x-1)^2=-3` `(x-1)=pmsqrt3i` `x=pmsqrt3i+1` `x=2(1/2pmsqrt3/2i)` `x-2(cospi/3pmisinpi/3)` `x=2e^(pmipi/3)` `alpha^n=2e^(ipi/3)`,`beta^n=2^n e^(-i npi/3)` `alpha^n+beta^n=2^n(cosnpi/3+isinnpi/3+cosnpi/3-isinnpi/3)` `=2^n2cos(npi/3)` `=2^(n+1) cos(npi/3)`.

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