All Questions
Tagged with sequences-and-series recurrence-relations
1,712
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Does this recurrent sequence have a limit?
I have a sequence $a_1 = 1$, $a_2 = 0$, $$a_k = \frac{k*(k+2)+(k+1)}{k*(k+2)}(a_{k-1} -\frac{k-1}{k*(k-2) + (k-1)}a_{k-2})$$ for $k \geq 3$,
I want to know whether this sequence would converge to $0$ ...
- 165
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1
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94
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IMO 2024 p-3,Sequence of Counts - Are Odd or Even Terms Eventually Periodic?
Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. We define $a_n$ for $n > N$ as the number of times $a_{n-1}$ appears in the list $a_1, ...
- 59
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Convergence question about $p_k=p_{k-1}^{p_0/(p_0+1)}+\ln(p_0)$ [closed]
Let $p_k\ge 2$ be a real number. For some $p_0$, consider the sequence
$p_1=p_0^{p_0/(p_0+1)}+\ln(p_0)$
$p_2=p_1^{p_0/(p_0+1)}+\ln(p_0)$
$p_3=p_2^{p_0/(p_0+1)}+\ln(p_0)$
$\vdots$
My question is how do ...
- 591
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173
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Prove or disprove the limit of a sequence is negative.
I have a sequence of positive numbers $\{f_k\}$ such that all the odd terms sum up to 1 and so do all the even terms, i.e. $\sum_{k=1}^{\infty}f_{2k-1}=\sum_{k=1}^{\infty}f_{2k}=1$, and $1>\sum_{i=...
- 165
3
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2
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181
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Let $a_1=2/5, a_{n+1}=a_n+a_n^2/n^2$, prove that $a_n<1$
$$
\mbox{Let}\qquad a_{1} = {2 \over 5},\quad a_{n + 1} = a_{n} + \frac{a_{n}^{2}}{n^{2}}
$$
Prove that
$a_{n} < 1,\ \forall\ \mbox{positive integers}\ n$.
I have tried to play around with the ...
- 1,727
4
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2
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84
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Getting explicit formula from non-homogenous recursive sequence
I am wondering whether there is an efficient way to find explicit formulas for non-homogenous recursive sequences.
While writing out the first few terms and ...
14
votes
4
answers
586
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Closed form: show that $\sum_{n=1}^\infty\frac{a_n}{(n+1)4^n}=\zeta(2) $
Let $a_n$ be the sequence defined via
$$ a_n=\sum_{k=1}^n{2n \choose {n-k}}\frac{1+(-1)^{k+1}}{k^2}$$
then prove that
$$\sum_{n=1}^\infty\frac{a_n}{(n+1)4^n}=\frac{\pi^2}{6} $$
I tried simplifying $...
- 3,392
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248
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how to find explicit formula for $f_{n+1}=af_n+\frac{b}{f_n}$?
I tried to find an explicit formula for the recurrence relation
$$f_{n+1}=af_n+\frac{b}{f_n} , f_0 \ne0$$
I will show what I got in five cases
Case1
for $f_0=-\frac{\sqrt{b}}{\sqrt{1-a}}\ne0$ I got ...
- 1,637
1
vote
2
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66
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Solve one conditional recurrence relation
Recently when learning SICP, Exercise 1.26 exposes one sequence when calculating its complexity (here it is the multiplication count):
$$
D(n)=
\begin{cases}
2D(n/2)+1,n\text{ is even}\\
...
- 416
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59
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How does $G_n$ relate to $F_n$ here? [duplicate]
Let $F_n$ be the $n$th Fibonacci number, i.e $$F_n=\left\{\begin{array}{cl}F_{n-1}+F_{n-2} & \text { if } n>1 \\ 1 & \text { if } n=1 \\ 0 & \text { if } n=0\end{array}\right.$$ and ...
- 3,019
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137
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$a_{n+1}=a_{n}+a_{n-1}+1$ relation with Fibonacci sequence
I am learning SICP. I am stuck at exercise-1.20
num_b(k+1) = num_b(k) + num_b(k-1) + 1
Obviously we could get
num_b(k) = fib(k+1) - 1
In the following, I will use latex to render the above equation
...
- 416
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Prove that the numbers 2008 and 2106 are not terms of this sequence.
The sequence $(x_n)$ is given recursively with $x_1 = 188$, and $x_{n+1}$ is obtained from $x_n$ by adding twice the sum of the digits of the number $x_n$.
Prove that the numbers 2008 and 2106 are not ...
user1316790
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3
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83
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Confused on iterative approach used in solving this recurrence relation.
Here they tried to solve the recurrence relation $a_{n} = n - 1 - a_{n-1}, a_{0} = 7$ using an iterative approach. I'm confused about the last two steps in this solution and how we went from the 2nd ...
- 883
-3
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63
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Let $\alpha, \beta$ be the roots of $x^2-a_1x+1=0$, and consider the sequence of numbers $a_r(r\ge0)$ with $a_0=1$ and $a^2_{r+1}=1+a_r.a_{r+2}$ [closed]
Let $\alpha, \beta$ are roots of equation $x^2-a_1 x+1=0$ and consider sequence of numbers $a_r,\;r\geq0\;$ with $a_0=1\;$ and $a_{r+1}^2=1+a_r\cdot a_{r+2}.\;$ Then which of the following is/are true?...
- 8,338
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71
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Finding a limit of a recurrently given sequence
I am struggling with one example of limit required for recurrently given sequence. There are 3 examples in total, I will also post the ones I did.
$ x_1>0$ and recursively given sequence: $$x_{n+1}...
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