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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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