9
votes
8answers
243 views

Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$

Question: Proof that if $$a_n=\left\{ \begin{array}{ll} a_1=1\\ a_{n+1}=1+\frac{1}{1+a_n} \end{array} \right.$$ then $a_n$ converge, and then find $\lim_{n \to \infty}a_n$. I found ...
26
votes
3answers
414 views

Finding every $n$ such that $a_n$ is an integer

Let us define $\{a_n\}$ as $a_1=a_2=1$,$$a_{n+2}=a_{n+1}+\frac{a_n}{2}\ \ (n=1,2,\cdots).$$ Then, is the following true? If $a_n$ is an integer, then $n\le 8$. I conjectured this by using ...
3
votes
1answer
63 views

Limit of a recursiv trigonometric function

So while doing my math homework I recently stumbled upon this little thing: It seems that $y_n = \sin(y_{n-1}) + k$ with arbitrary $y_0$ and $k$ converges to a definite value for $n \to \infty $. ...
0
votes
0answers
43 views

How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0 $ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...
0
votes
0answers
30 views

Standard results for limit of recurrence relation?

Is there a standard result out there that gives the following? It looks graphically like it must be true, but I'd like to appeal to a known result if possible. Let $f:[0,1]\to[0,1]$ be a continuous, ...
2
votes
2answers
92 views

How to solve the recurrence $T(n) = T(n-1) +\sqrt{n}$?

While solving the recurrence of the title I come to the series $$T(n) = \sqrt1 + \sqrt2 + \sqrt{3} + \cdots + \sqrt n.$$ Please somebody help me how to solve this.
0
votes
1answer
29 views

Solve the recurrence relation

Assuming that $n$ is a power of $2$, solve the recurrence relation $$T(n)=2T\left(\frac{n}{2}\right)+2$$ Take $T(2)=1$ and $T(1)=0$. Also how can this be done with the master theorem, if possible?
5
votes
2answers
355 views

Series of sequence converges?

Given the recursively defined sequence $$ a_2 = 2(C+1)a_0 $$ and $$ a_{n+2}=\frac{\left(n(n+6)+4(C+1)\right)a_n - 8(n-2)a_{n-2}}{(n+1)(n+2)}. $$ that I got from the Frobenius method applied to an ODE, ...
1
vote
1answer
61 views

Concerning a specific family of recursive sequences

There is a family of recursive sequences that I came up with: $$a_n=\text{sum of factors less than $a_{n-1}$ of } a_{n-1}, a_0\in\mathbb{N}$$ First question: Has it been studied before? Here are ...
3
votes
2answers
72 views

Find the limit of the sequence given by recurrence relation

Find the limit of the sequence given by recurrence relation $c_{n+1} = (1-\frac{1}{n})c_n + \beta_n $ where $ \beta_n $ is any sequence with the property $n^2\beta_n \to 0$ as $ n \to \infty$ I've ...
1
vote
0answers
67 views

Recurrence Relation Challenges

as i ask question and answered by some Clever people at this topic: Recurrence Relation Solving Problem i try to learn new thing with new question very similar to get familiar with recurrence ...
2
votes
1answer
64 views

Generating functions over $\mathbb{Z}$

Let $(a_n)_{n \in \mathbb{Z}}$ be a sequence such that both limits $\lim_{n \to \infty} a_n$ and $\lim_{n \to -\infty} a_n$ exists. Consider recursive relation $$ 2b_n - \frac{1}{2}(b_{n-1} + b_{n+1}) ...
1
vote
0answers
29 views

Finite geometric sequence with a ratio greater than 1

I am trying to solve the recurrence relation: $ t(n)= 3T(n/2)+Cn$ (if $n>2$) $t(2)=C$ (otherwise) I know that there is the Master Theorem but I am trying to use the tree method. This ...
0
votes
1answer
41 views

Problem with making explicit formula from recursive formula

I'm having trouble turning this recursive formula into explicit one $a_0 = 0,\\a_{n+1} = a_n(1 - b_n) + b_n,$ where $(b_n)$ is given sequence of real numbers.
0
votes
1answer
66 views

Understanding Recurrence Relation

as i ask question and answered by some Clever people at this topic: Recurrence Relation Solving Problem i try to learn new thing with new question very similar to get familiar with recurrence ...
1
vote
2answers
73 views

Find the general term of the recursive relation $a_{n+1}=\frac{1}{n+1}\sum\limits_{k=0}^n a_k 2^{2n-2k+1}$

I need to find the general term of the recursive relation $a_{n+1}=\frac{1}{n+1}\sum\limits_{k=0}^n a_k 2^{2n-2k+1}$ I know it's the central binomial sequence but I can't find a way to show it. ...
1
vote
4answers
65 views

A sequence was defined by a equation

The sequence was defined by the equations: $${a}_{1}=a\left(\in R \right),{a}_{n+1}=\frac{2{{a}_{n}}^{3}}{1+{{a}_{n}}^{4}},n\geq 1.$$ show that $\left(a \right)$The given sequence is convergent. ...
0
votes
2answers
43 views

Closed form for a strong recurrence relation

Let $\alpha_n$ be a sequence of complex numbers and consider the sequence $b_n$ defined by the (strong) recurrence relation : $$b_{n+1} = \sum_{k=0}^n \alpha_{n-k} b_k$$ with the initial condition ...
4
votes
4answers
242 views

Finding the billionth number in the series: $2, 3, 4, 6, 9, 13, 19, 28, 42, \ldots $?

Series is defined as $$a_{n+1} = \lfloor\frac{3\cdot a_n}{2}\rfloor,\qquad a_0 = 2$$ It can be viewed as the number of animals starting from a single pair if any pair of animals can produce a single ...
4
votes
1answer
94 views

Is there a generating function for $\sqrt{n}$?

I tried to come up with a closed form for the ordinary generating function for the sequence $\{\sqrt{n}\}_0^{\infty}$ but I could not. Is there a way to derive it using the recurrence relation ...
1
vote
2answers
74 views

How many $a$-nary sequences of length $b$ never have $c$ consecutive occurrences of a digit?

Let $S(a,b,c): = \#\{a$-nary sequences of length $b$ without $c$ consecutive occurrences of a digit$\}$. For example, $S(2,n,3)$ would be the number of binary sequences of length $n$ without $3$ ...
5
votes
0answers
34 views

Finding a recurrence relation for a sequence not containing $0$ in the digits. Also showing $\sum\limits_{k=1}^n \dfrac{1}{a_k} < 90$

Let $1,2,3,4,5,6,7,8,9,11,12,\ldots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that ...
0
votes
1answer
23 views

Going from recurrence relations to closed form

How do I go from the following recurrence relation a(n) = (n+1)a(n-1) where a(0) = 2 to a closed form? I know I need to use an iterative approach but I am not ...
3
votes
2answers
90 views

Proving integrality of a sequence of numbers

How would one go about proving whether or not the terms of a sequence such as the following $$1, 1, 1, 1, 1, 6, 21, 181, 5221, 1090981, 986241401, 51490676208426,\ldots$$ defined by ...
0
votes
1answer
27 views

Relation between coefficients of Maclaurin series and recurrence relations

Suppose $I_{2n}=\int_0^{\pi}x\sin^{2n}x \;dx$. Then one can obtain the following: $$I_0= \frac{\pi ^2}{2}, I_1= \frac{\pi ^2}{4}, I_2= \frac{3\pi ^2}{16}, I_3= \frac{5\pi ^2}{32},\ldots, I_n= ...
0
votes
1answer
74 views

Induction Proof of: Find $f(n)$ when $n=2^k$, where $f$ satisfies the recurrence relation $f(n)=f(\frac{n}2)+1$ with $f(1)=1$

How can I proof this using regular induction and strong induction? $$f(2^k)=f(\frac{2^k}2)+1=f(2^{k-1})+1$$ $$f(2^{1-1})=f(2^0)=f(1)=1$$ $$f(2^{2-1})=f(2^1)=f(2)=f(1)+1=2$$ ...
1
vote
2answers
77 views

Get the Nth term of a sequence 1,2,4,7,13,24…

I have a sequence: 1,2,4,7,13,24,44,81, ... and I think it's like a Fibonacci sequence, however you add three number together and not two ("Tribonacci"?). So: $$ v_n = v_{n-1} + v_{n-2} + v_{n-3} $$ ...
0
votes
1answer
25 views

help me finding the limit of sequence question

A 1= 1 and {A n}=root[1 + {A n-1}] B 1= 2 and {B n}=root[2*{B n-1}] help me Since I am studying math recently I need person`s help
0
votes
0answers
36 views

Total Time for a ball that bounces from a height of 8 feet and rebounds to a height 5/8 [duplicate]

I am trying to solve the following problem. Let's say a ball is dropped from a height of $8$ feet and rebounds to a height $\frac{5}{8}$ of its previous height at each bounce keeps bouncing ...
0
votes
2answers
80 views

Total Time a ball bounces for from a height of 8 feet and rebounds to a height 5/8

I am trying to solve the following problem. Let's say a ball is dropped from a height of $8$ feet and rebounds to a height $\frac{5}{8}$ of its previous height at each bounce keeps bouncing ...
0
votes
2answers
39 views

Solve Fibonacci-like linear recurrence equation

How to solve the following equation: $f(n) = f(n-1) + f(n-2) + 1$ My best guess is that it has something to do with Linear Recurrence Equation. I know how to do it without the constant $1$, which ...
0
votes
2answers
45 views

If infinitely many points of a sequence are given, is it possible to find out the recurrence relation?

I was thinking about sequences and stumbled upon a concept. If infinitely many points of a sequence are given, is it possible to find out the recurrence relation? Please enlighten.
1
vote
3answers
35 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 ...
3
votes
2answers
80 views

Fibonacci General Formula - Is it obvious that the general term is an integer? [duplicate]

Given the recurrence relation for the Fibonacci numbers, $F_{n+1}=F_{n}+F_{n-1}$ with $F_0=1$ and $F_1=1$ it's obvious that $F_n$ is a positive integer for all $n$. Suppose instead we were given ...
2
votes
2answers
35 views

A mixture of AP and GP

A battery loses $10$ mAh of after every hour when in use. The same battery loses $1\%$ of its current amount of charge every hour when not in use. Suppose that the battery is fully charged with ...
1
vote
0answers
41 views

Monotonicity of a recursively defined sequence $s_{n+1} = \sqrt{4s_n -1}$

I am trying to answer this question: Let $s_1 = k$ and $s_{n+1} = \sqrt{4s_n -1}$. For what values of $k$ will the sequence $s_n$ be monotone increasing? I know the definition of monotone increasing ...
3
votes
1answer
168 views

Estimating linearly independent solutions to third order recurrence relation

I'm trying to prove something about two linearly independent solutions; $a_n$, $b_n$, to a recurrence relation I have - specifically that $\left| \frac{a_n}{b_n} \right|$ is eventually monotonically ...
6
votes
3answers
323 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
6
votes
2answers
496 views

Recursive square root problem [duplicate]

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 $$ ...
0
votes
2answers
49 views

Show that there is a unique sequence of positive integers $(a_n)$ satisfying $a_1=1,a_2=2,a_4=12,a_{n+1}a_{n-1}=a_n^2\pm 1 $

Show that there is a unique sequence of positive integers $(a_n)$ satisfying the following conditions. $$a_1=1,a_2=2,a_4=12,a_{n+1}a_{n-1}=a_n^2\pm 1$$ I approached the problem to find out, ...
9
votes
1answer
292 views

Prove that this particular sequence contains an infinite number of sixes

Given the sequence $$2,7,1,4,7,4,2,8,\ldots$$ which begins with $2, 7$ and is constructed by multiplying successive pairs of its members and adjoining the results as the next one or two members of ...
0
votes
2answers
97 views

Does this series $2 + 4 + \cdots + \sqrt{\sqrt{n}} + \sqrt{n} + n$ have a general term?

Does this sum simplify to a general term in terms of $n$? If so, how would you arrive at that term? $2 + 4 + \cdots + \sqrt{\sqrt{n}} + \sqrt{n} + n$. Thanks.
0
votes
1answer
49 views

Elementary Functions Name: f(a,b) = f(b,a-1)+b

I am quite simply looking for a function that I forgot about from way back when. I am positive I learned this at some point in grade school, but I just can't remember what it is called! The function ...
3
votes
3answers
140 views

Proof Using the Monotone Convergence Theorem for the sequence $a_{n+1} = \sqrt{4 + a_n}$

Consider the recursively defined sequence $a_0 = 1$ $a_{n+1} = \sqrt{4 + a_n}$ How to prove that the sequence converges using the Monotone Convergence Theorem, and find the limit?
2
votes
1answer
53 views

Generating Function for Recurrence Relation in 2 Variable

I have a recurrence relation with 2 variables similar to $$ F(n,m) = n\cdot F(n-1,m) + (n-m)\cdot F(n-1,m-1) $$ I want to know the steps required to get the generating Function for such recurences. I ...
6
votes
6answers
505 views

How to find the limit of this recurrence relation?

$a_n$ 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 ...
1
vote
3answers
85 views

How to go about solving this recurrence relation?

In my discrete math class we were given the problem of finding an explicit formula for this recurrence relation and proving its correctness via induction. $$a_n=2a_{n-1}+a_{n-2}+1, a_1 = 1, a_2=1.$$ I ...
0
votes
0answers
40 views

Removing the Summation (Closed Form)

The following question from "Combinatorics of Permutations" : $$ E[X] = \sum\limits_{k = 2}^n \frac{k\cdot T(n,k)}{n!} $$ where $$ T(n,k) = k \cdot T(n-1, k) + 2 \cdot T(n-1, k-1) + (n-k) \cdot ...
0
votes
0answers
18 views

How to solve this recurrence relation consisting of 9 equations (one of them with a minimum function)?

Given the recurrence relation as follows. $T_{k}=min((W_{k}+Y_{k}) , (X_{k}+Z_{k}))$ $A_{k}=T_{k}\frac{W_{k}}{W_{k}+Y_{k}}\frac{X_{k}}{X_{k}+Z_{k}}$ ...
2
votes
2answers
55 views

Got stuck in proving monotonic increasing of a recurrence sequence

I'm stuck with the prove of the following recurrence sequence which was part of and old exam. $a_1:=\frac{1}{2}, a_{n+1}:=a_n(2-a_n)$ for $n \in \mathbb{N}$ I have to show that $0 <a_n < 1$ ...