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Finite sequence


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For sequence'

a n = a n-1 + a n-2

We first assume that a n has a general form of b^n.

Then, we put it into the sequence equation and get a quadratic one.

It will be ok for me if the final solution for the a n be that form.

But normally, if there exists two different roots, the final solution indeed is in the form of (C b^n + D f^n)

where they are arbitrarily assigned.

Why do we assume the incorrect form but getting the correct answer?

Any comments Please!

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We do not assume that the solution is b^n at all. We assume that the solution is of the form c_1*b_1^n + c_2*b_2^n: the equation is linear. And then from this knowledge we know that b must satisfy a quadratic equation based upon the information given. All you're doing is taking sloppy shorthand for absolute truth.

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*may* have solutions of the form, not *does* have. In anycase, it is badly written, but that is not uncommon in textbooks (it for instance states the solution in the degree one case is a^nx_0 and then immediately asserts x_n=a^n is a solution. It is not unless x_0=1

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