Linked Questions

14
votes
6answers
1k views

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 ...
3
votes
2answers
2k views

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: $P_{0} = 0$ $P_{1} = 1$ ...
4
votes
1answer
1k views

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 ...
2
votes
3answers
266 views

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 ...
3
votes
4answers
264 views

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 ...
1
vote
3answers
197 views

How to deal with linear recurrence that it's characteristic polynomial has multiple roots?

example , $$ a_n=6a_{n-1}-9a_{n-2},a_0=0,a_1=1 $$ what is the $a_n$? In fact, I want to know there are any way to deal with this situation.
3
votes
2answers
234 views

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: $$1+5u_nu_{n+1}=k^2,\; k \in \mathbb N.$$
1
vote
3answers
247 views

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 ...
0
votes
2answers
263 views

How does one rewrite a recursive function to be strictly non-recursive?

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 ...
0
votes
1answer
243 views

Explicit formula for $a_n$, reccurence relations

For the following, solve each of the following recurrence relations by giving explicit formula for $a_n$ and calculate $a_9$. $a_n = 10 a_{n-1}, a_0 = 3; $ $a_n = -a_{n-1}, a_0 = 5;$ $a_n = 3 a_{n-1} ...
1
vote
1answer
218 views

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 ...
1
vote
3answers
112 views

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 ...
1
vote
3answers
40 views

Finding the limit of a recurrence relation?

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 ...