# Tagged Questions

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### Proof of convergence of $a_{n+1} = \dfrac{a_n^2 + 1}{3}$ in $\mathbb{R}$ and finding its limit

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
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### What is the closed for of this sequence? [closed]

How can I find the closed form of the sequence k generated by input n? \begin{array}{c|c} n & k\\\hline 1 & 0\\ 2 & ...
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### Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...
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### How prove there exist postive integer $n$ such $x_{n}>y_{n}$

let two positive sequence $\begin{cases} x_{n+2}=x_{n}+x^2_{n+1}\\ y_{n+2}=y^2_{n}+y_{n+1} \end{cases}$ and $x_{1}>1,y_{1}>1,x_{2}>1,y_{2}>1$ show that: there exists $n$, such ...
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### Two sequences defined by recurrence relations satisfy $x_n/y_n<\sqrt{7}$ for all $n$

Let $(x_n)_{n\geq 1}$ and $(y_n)_{n\geq 1}$ be two sequences such that: $$x_{n+1}=x_n^2+1 \quad \text{ and } \quad y_{n+1}=x_n y_n$$ with $x_1=2$ and $y_1=1$ Prove that for all $n$ ...
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### 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 ...
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### 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.
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### 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 ...
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### 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. ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $3$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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Suppose $I_{2n}=\int_0^{\pi}x\sin^{2n}x \;dx$. Then one can obtain the following: $$I_0= \frac{\pi ^2}{2}, I_1= \frac{\pi ^2}{4}, I_2= \frac{3\pi ^2}{16}, I_3= \frac{5\pi ^2}{32},\ldots, I_n= ... 1answer 77 views ### Induction Proof of: Find f(n) when n=2^k, where f satisfies the recurrence relation f(n)=f(\frac{n}2)+1 with f(1)=1 How can I proof this using regular induction and strong induction?$$f(2^k)=f(\frac{2^k}2)+1=f(2^{k-1})+1f(2^{1-1})=f(2^0)=f(1)=1f(2^{2-1})=f(2^1)=f(2)=f(1)+1=2$$... 2answers 82 views ### Get the Nth term of a sequence 1,2,4,7,13,24… I have a sequence: 1,2,4,7,13,24,44,81, ... and I think it's like a Fibonacci sequence, however you add three number together and not two ("Tribonacci"?). So:$$ v_n = v_{n-1} + v_{n-2} + v_{n-3}  ...
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 ...