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If e and f are the roots of the equation ax^2+bx+c=0 and sn=e^n+f^n show that (asn+1)+bsn+csn-1=0 and hence find s5

If e and f are the roots of the equation ax^2+bx+c=0 and sn=e^n+f^n show that (asn+1)+bsn+csn-1=0 and hence find s5

Grade:11

1 Answers

shaswat roy
19 Points
9 years ago

well im assuming what you meant was to prove:  a*s(n+1) + b*s(n) + c*s(n-1) = 0

this implies, we have to prove   ae^(n+1) + af^(n+1) + be^n + bf^n + ce^(n-1) + cf^(n-1) = 0

from the terms containing powers of e take e^(n-1) common, same for the terms containing f.

so we get  [e^(n-1)][ ae^2 + be + c] + [f^(n-1)][ af^2 + bf + c]

since e and f are roots of the given quadratic eqn, the above expression becomes 0.

to find s(5) plug in values of n and use the relations e+f=-b/a and ef=c/a

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