InterviewSolution
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InterviewSolution
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If `alpha`, `beta`, `gamma` are such that `alpha+beta+gamma=2`, `alpha^2+beta^2+gamma^2=6`, `alpha^3+beta^3+gamma^3=8`, then `alpha^4+beta^4+gamma^4` |
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Answer» We have, `(alpha+beta+gamma)^(2)=alpha^(2)+beta^(2)+gamma^(2)+2(betagamma+gammaalpha+alphabeta)` `rArr 4 = 6 + 2 (betagamma+gammaalpha+alphabeta)` `rArr betagamma+gammaalpha+alphabeta = -1` Also, `alpha^(3) + beta^(3) + gamma^(3) - 3alphabeta gamma` `=(alpha+beta+gamma)(alpha^(2)+beta^(2)+gamma^(2))(betagamma+gammaalpha+alphabeta)` or " " 8 - 3 `alpha+beta+gamma = 2(6+1)` or `3alpha+beta+gamma = 8 - 14 = -6` or `alpha+beta+gamma = - 2` Now, `(alpha^(2)+beta^(2)+gamma^(2))^(2)=Sigmaalpha^(4)+2Sigmaalpha^(2)beta^(2)` `=Sigmaalpha^(4)+2[(Sigmaalphabeta)^(2)-2alphabetagamma(Sigmaalpha)]` or `=Sigmaalpha^(4)= 36 - 2 [(-1)^(2)-2(-2)(2)] = 18` |
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