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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: $P_{0} = 0$ $P_{1} = 1$ ...
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### Solving a recurrence relation with the characteristic equation

I have some trouble solving this due to not seeing the steps to be able to feed it into the characteristic equation. $$T(n) = 4T(n-2) +n + 2^nn^2\ \text{with}\ \ T(0)=0,\ T(1)=1$$ (don't have to ...
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### Closed Form of Recurrence Relation

I have a recurrence relation defined as: $$f(k)=\frac{[f(k-1)]^2}{f(k-2)}$$ Wolfram Alpha shows that the closed form for this relation is: $$f(k)=\exp{(c_2k+c_1)}$$ I'm not really sure how to go ...
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### If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent.

If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent. What is the limit? The back of my textbook says that ...
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### Find n that : $1+5u_nu_{n+1}=k^2, k \in N$

Let ${u_n}$ be such that: $$\begin{cases}u_1=20;\\u_2=30;\\ u_{n+2}=3u_{n+1}-u_{n},\; n \in \mathbb N^*.\end{cases}$$ Find $n$ such that: $$1+5u_nu_{n+1}=k^2,\; k \in \mathbb N.$$
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### Solution to a linear recurrence

What is the general solution to the recurrence: $x(n + 2) = 6x(n + 1) - 9x(n)$ for $n \geq 0$; with $x(0) = 0; x(1) = 1$? Solution. The first few values of $x(n)$ are $0,1,6,27,...$ The auxiliary ...
Given the recursive function: $$f(0) = \frac{x^2}{2} + \frac{x}{2}, f(n) = \frac{f(n-1)}{2} + \frac{x}{2}$$ where $x$ = some integer How would one rewrite this function to be strictly ...