# Tagged Questions

Questions regarding functions defined recursively, such as the Fibonacci sequence.

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### Is the sequence $x_{n+1} = \frac{x_n + \alpha}{x_n + 1}$ convergent?

Fix $\alpha > 1$, and consider the sequence $(x_n)_n \geq 0$ defined by $x_0 > \sqrt \alpha$ and $$x_{n+1} = \frac{x_n + \alpha}{x_n + 1}, n = 0, 1, 2, \dots$$ Does this sequence converge, ...
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### How to solve this recurrence $K(n)=2K(n-1)-K(n-2)+C$?

The recurrence is $K(n)=2K(n-1)-K(n-2)+C$ where $C$ is a constant. What I have tried is substituting $2K(n-1)$ as we do in fibonnacical recurrences. It didn't gave me a fruitful expression! Can ...
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### How to find $\lim a_n$ if $a_{n+1}=a_n+\frac{a_n-1}{n^2-1}$ for every $n\ge2$

$a_n$ is a sequence where $a_1=0$ and $a_2=100$, and for $n \geq 2$: $$a_{n+1}=a_n+\frac{a_n-1}{(n)^2-1}$$ I have a basic understanding of sequences. I wasn't sure how to deal with this recurrence ...
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Give a precise meaning to evaluate the following: $$\large{\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dotsb}}}}}$$ Since I think it has a recursive structure (does it?), I reduce the equation to $$p=\... 4answers 150 views ### closed form for d(4)=2, d(n+1)=d(n)+n-1? I am helping a friend in his last year of high school with his math class. They are studying recurrences and proof by inference. One of the exercises was simply "How many diagonals does a regular n-... 5answers 263 views ### All the ternary n-words with an even sum of digits and a zero. I'm trying to find a recursive formula for all the ternary (using {0,1,2}) sequences of length n which contain at least one zero, and have an even sum of digits. My attempt so far is added below. ... 5answers 130 views ### Recurrence of the form 2f(n) = f(n+1)+f(n-1)+3 Can anyone suggest a shortcut to solving recurrences of the form, for example: 2f(n) = f(n+1)+f(n-1)+3, with f(1)=f(-1)=0 Sure, the homogenous solution can be solved by looking at the ... 3answers 151 views ### Prove that A_{100} \gt 14 where A_{n}=A_{n-1}+\frac{1}{A_{n-1}} and A_1=1 I tried attempting the question, and the best upper bound I could obtain was 1+\ln{98}. I tried using A_{n}\le n to form a harmonic series, but that wasn't strong enough. Any help would be ... 2answers 79 views ### How to evaluate \lim_{ n\to \infty }\frac{a_n}{2^{n-1}}, if a_0=0 and a_{n+1}=a_n+\sqrt{a_n^2+1}? Let a_1,a_2,..,a_n be sequence of real numbers such that a_{n+1}=a_{n}+\sqrt{1+a_n^2} and a_0=0. How to evaluate \lim_{ n\to \infty }\frac{a_n}{2^{n-1}} ? 2answers 239 views ### A recurrence relation problem: \frac {a_{n-1}.a_{n+1}} {a_n^2} = 1 + \frac 1 n I need to solve this recurrence problem to find a_n \dfrac {a_{n-1}.a_{n+1}} {a_n^2} = 1 + \dfrac 1 n It is what I tried so far:$$\log (\dfrac {a_{n-1}.a_{n+1}} {a_n^2}) = \log(1 + \dfrac 1 n)$... 1answer 209 views ### How to solve this recurrence relation$a_{n+2} = \sqrt{a_{n+1}\cdot a_{n}}$? I am trying to solve the recurrence: $$a_{n+2} = \sqrt{a_{n+1}\cdot a_{n}}$$ but here is a problem for me. After few steps I have this: $$a_n^2 = a_{n-1}\cdot a_{n-2}$$ and I don't now what to do ... 5answers 160 views ###$x_{n+1} = \sqrt{3x_n}$converges for$x_1 = 1,x_1 = 27$Proof Prove$x_{n+1} = \sqrt{3x_n}$converges for$x_1 = 1,x_1 = 27$. (Separate problems for$x_1 = 1$and$x_1 = 27$.) EDIT: Took out bad algebra. 5answers 171 views ### Convergence of a recurrence Given the recursive definition (starting with a positive integer) $$a_n = \frac{a_{n-1}}{2}+4$$ I am trying to find an explicit form and show that it approaches 8. So I started by writing it out, ... 3answers 93 views ### Please solve this recurrence relation question for$8a_na_{n+1}-16a_{n+1}+2a_n+5=0$Suppose$a_1=1$and $$8a_na_{n+1}-16a_{n+1}+2a_n+5=0,\forall n\geq1,$$Please help to sort out the general form of$a_n$. Here are the first a few values of the series. Not sure if they are useful as ... 4answers 291 views ### Alternating Recurrence relation$a_n = b_{n-1} + 5$and$b_n = na_{n-1}$I am racking my brain on solving the relation where: $$a_n = b_{n-1} + 5$$ $$b_n = na_{n-1}$$ where$a_0$=$b_0$= 1 I am trying to find the closed form for$a_n$. I have tried to shifting$b_n = ...
I'm trying to find the general term of the recurrence relations $\quad a_{n+1}=a_n+\text hb_n$ $\quad b_{n+1}=b_n-\text ha_n$ $\quad a_0=0, \quad b_0=1$ I tried finding the terms, \$...