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Understanding Series stuff

Now i have managed to solve the attached problem using some instructions given to me. But i really do not understand how this works.

Under what premise can I assume that $\displaystyle a_{n}=t^n$ then I solve the equation $\displaystyle t^n = 3t^{n-1}-t^{n-2}$. then I was told that $\displaystyle a_{n}$ is a linear combination of the roots.

I solve this question fine, I just really do not understand any of the theory.

Could someone show me some reading material or a proof of this method.

Many Thanks

Bobak