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### How to solve this recurrence relation? $f_n = 3f_{n-1} + 12(-1)^n$

How to solve this particular recurrence relation ? $$f_n = 3f_{n-1} + 12(-1)^n,\quad f_1 = 0$$ such that $f_2 = 12, f_3 = 24$ and so on. I tried out a lot but due to $(-1)^n$ I am not able to ...
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### Finding the closed form for a sequence

My teacher isn't great with explaining his work and the book we have doesn't cover anything like this. He wants us to find a closed form for the sequence defined by: $P_{0} = 0$ $P_{1} = 1$ $\vdots$...
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### Limit of a contractive sequence

Given: $a < b < 0$ and $y_1 = a$ $y_2 = b$ $y_n = \frac{1}{3}y_{n-1} + \frac{2}{3}y_{n-2}$, for $n > 2$ I was able to show that this sequence was contractive and now I'm asked to find the ...
### A sequence polynomial $P_n(x)$
Given the polynomial sequence $(P_n(x))$ satisfying $$P_0(x)=P_1(x)=1$$ $$P_{n+2}(x)=P_{n+1}(x)+xP_n(x)$$ Find $P_n(x)$ I know $P_n(x)=\sum_{k\ge 0} {n-k\choose k}x^k$ but don't know how to solve ...
I have a sequence defined by the relation $$x_{n+1} = \alpha x_n + (1-\alpha)x_{n-1}$$ for $n\geq 1$, and I want to find the limit in terms of $\alpha , x_0,x_1$. I tried to do this by setting up a ...