1
vote
2answers
17 views

Recurrence problem with a game of probability [duplicate]

Fair coin flipping (50% on both sides) $P_1$ and $P_2$ plays a few games of fair coin flipping. Assume player $A$ starts with $x$ coins and player $B$ with $y$ coins. Let $P_n$ denote the ...
2
votes
5answers
24 views

Solve recursion with constant added

I have the following problem: Define a sequence $(a_n)$ where $a_1 = 4$ and $a_n = 4a_{n-1} - 4$. Find a closed form for $a_n$. So basically I usually know how to deal with recursions like $a_n = ...
1
vote
0answers
42 views
+50

Recurrence vs Recursive

Say team 1 is studying the recursive characteristics of a function. Team 2 is studying the recurrent characteristics of the same function. Are the 2 teams studying the same thing? I have found for ...
0
votes
1answer
39 views

Find Recurrence Relation of Code

Suppose A(n) be the number of stars that wrote with the following example. for n>=3, i want calculate the recurrence relation for this code. any idea or solution? ...
1
vote
1answer
21 views

Time Complexity of one Challenging Example

Anyone would help me to calculate the order (time complexity) of this example ?
1
vote
1answer
39 views

Number of strings of size $k$ that do not have 'ab'

Consider $\Sigma = \{a,b,c\}$ and the language $L$, the set of all strings that do not contain 'ab' Find strings, of size $k$ is in $L$ ($L_k$) Consider $A_k$ (strings of size $k$ that end in $a$) ...
1
vote
1answer
46 views

Problematic Initial Condition of a Recurrence Relation

I encountered this equation, and tried to solve it: $T(n) = T(\sqrt{n})+log(n)$ Under the initial condition T(1)=1. Can someone tell me why is this initial condition helpful? I mean, of course ...
2
votes
1answer
31 views

Recurrence relation from code

My Friends, Hi, I see an old book in mathematics for computer science. everyone could help me, for example how we calculate the order (Time complexity) of following code: ...
1
vote
2answers
56 views

Solving a recurrence relation with square root

I ran into a bad recurrence relation. Anyone would calculate T(n) or add some hint? $$T(n) = \begin{cases} n,\quad &\text{ if n=1 or n=0 }\\ \sqrt{1/2[T^2(n-1)+T^2(n-2)]+}n ,\quad ...
2
votes
1answer
54 views

Height of Recursion Tree for $T(n, k) = T(n/2, k) + T(n, k/4) + kn$

If we use a recursion tree for solving $$\begin{cases} T(*,1)=T(1,*)=a \\ T(n,k)=T(n/2,k)+T(n,k/4)+kn \end{cases}$$ What is the height of the recursion tree? Any idea or solution highly ...
0
votes
1answer
70 views

Solving the recurrence relation $T(n)=4T\left(\frac{\sqrt{n}}{3}\right)+ \log^2n$

How we calculate the answer of following recurrence? $$T(n)=4T\left(\frac{\sqrt{n}}{3}\right)+ \log^2n.$$ Any nice solution would be highly appreciated. My solution is: $n=3^m \to ...
3
votes
0answers
39 views

Solving the recurrence $T(n)=4T(\frac{\sqrt{n}}{3})+ \log^2n$ [closed]

How we calculate the answer of following recurrence? $$T(n)=4T\left(\frac{\sqrt{n}}{3}\right)+ \log^2n.$$ Any nice solution would be highly appreciated.
3
votes
2answers
95 views

If a recursive sequence converges, must its inverse be divergent?

Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = ...
2
votes
2answers
37 views

Solving a Recursive equation, wrong options? [closed]

This is a very simple recursive formula. Are the options provided wrong? $$ T(n) = T\left(\frac{n}2\right) + 2 $$ with $T(1) = 1$. What's the solution when $n=2^k$ ($n$ is a power of $2$)? The ...
0
votes
3answers
56 views

Finding general formula for a recursion function

If we have a recursion relation defined as $a_n = 3a_{n-1}+1$ with $a_1=1$ then find the general formula for $a_n$ in terms of $n$ with a(1) = 1. So far I have: $a_n = 3a_{n-2}+1+1 = 3a_{n-3}+1+1+1 ...
0
votes
1answer
59 views

Recursive Function - $f(n)=f(an)+f(bn)+n$

I've got this recursive function $f(n)=f(an)+f(bn)+n$, and I need to find $θ$ on $f(n)$, as $a+b>1$. Using a recursive tree, I managed to bound it by $n\log(n)$ from the bottom and by $n^2$ from ...
1
vote
2answers
137 views

Find an explicit formula for the recursive sequence (tips?)

Problem: A sequence is defined recursively as follows: Sk = 2k - Sk - 1, for all integers k greater than or equal to 1 S0 = 1 Use iteration to guess the explicit formula for the sequence. Use ...
2
votes
2answers
53 views

Recurrence relation practice problem that I can't figure out

Thanks for taking the time to look at this problem. I'm trying to prepare for a test on Monday by doing some extra odd numbered problems from my textbook. I'm having a lot of trouble trying to solve ...
0
votes
2answers
49 views

Recurrence Relationship Questions

Consider the recurrence defined by: $$G_0 = 0\\ G_n = G_{n-1} + 2n - 1$$ Determine what Gn is for several values of n to determine a formula for Gn. $2n$ $n$ $2n-1$ $n^2$ *I believe this one is ...
1
vote
1answer
45 views

Simple Recurrence Questions [closed]

$r$ is a real number, define a recurrence relationship for $A_n$ $$A_0 = 1\\ A_n = r\cdot A_{n-1}$$ Question: What is the value of $A_4$ $4(A{n-1})$ $r^4$ $1$ $4r$ I've pretty much eliminated ...
2
votes
1answer
145 views

Guess an explicit formula for recursively defined sequence

Was given this as a question. "Use iteration to guess an explicit formula for the recursively defined sequence and then prove that the formula is a solution to the recurrence using induction: ...
2
votes
5answers
93 views

How to find first five terms of sequence?

I'm new to recursion so please bear with me. I have to find the first five terms of a sequence with initial conditions $u_1 = 1$ and $u_2 = 5$, and, for $n \geq 3$, $$u_n = 5u_{n−1} − 6u_{n−2}.$$ I ...
1
vote
3answers
171 views

Recurrence relations (Big-O notation)

Say I'm given a recursive function such as: function(n) { if (n <= 1) return; int i; for(i = 0; i < n; i++) { function(0.8n) } } ...
0
votes
3answers
34 views

A closed form for the recursion?

Let $x$ and $y$ be real numbers and $x < y$ Given the recursion: $m_0 = \frac{x+y}{2}$ and $m_1 =\frac{m_0+ y}{2}$, so in general, $$m_i = \frac{m_{i-1} + y}{2}$$.. What is $m_{\infty}$? thanks ...
2
votes
1answer
80 views

Solve the recurrence relation:$T(n)=\sqrt{n}T\left(\sqrt{n}\right)+\sqrt{n}$ [closed]

I have doubt in solving the following questions: $T(n)=2T(\sqrt{n})+n$ $T(n)=\sqrt{n}T(\sqrt{n})+c$ $T(n)=\sqrt{n}T(\sqrt{n})+\sqrt{n}$ T(2)=1 for all the problems Atleast give the final answer.
1
vote
1answer
62 views

Creating Recurrence

If I have an integer $n \geq1$, and I had to draw $n$ straight lines, so that no two of them are parallel as well as no three of them intersect in one single point. These lines divide the plane into ...
3
votes
1answer
143 views

Solve Recurrence Equation with Induction

Question: Given the recurrence equation for the recursive Fibonacci sequence program: $T(n) = T(n-1) + T(n-2) + b$ $T(0) = T(1) = a$ Using induction, show that $T(n) \leq f(n)$, where $f(n) = c2^n, ...
0
votes
1answer
46 views

Master's theorem applicability.

I have to find out if the following recurrence can be solved with the master theorem: $$T(n) = 3T\left(\frac{n}{2}\right) + n^{\log\log n}$$ I have figured that, here, I have the third case because ...
4
votes
0answers
157 views

General overview about Recursion (free online texts)

I'm looking for free online texts about recursion. What I'm looking for formal definitions* of "all" (most of) the different types of recursion and from different points of views like Category ...
3
votes
4answers
137 views

Proving convergence of a sequence

Let the following recursively defined sequence: $a_{n+1}=\frac{1}{2} a_n +2,$ $a_1=\dfrac{1}{2}$. Prove that $a_n$ converges to 4 by subtracting 4 from both sides. When I do that, I get: ...
0
votes
1answer
41 views

Recurrence relation that skips $n-1$?

A difficult question I'm having problems with regarding recurrence relations and recurrence equations. The question is as follows: In how many different ways can I cover a 3xn checkberboard with 2x1 ...
0
votes
3answers
166 views

Difficult recursion problem

A student can do the things bellow: a. Do his homework in 2 days b. Write a poem in 2 days c. Go on a trip for 2 days d. Study for exams for 1 day e. Play pc games for 1 day A schedule of n days ...
2
votes
4answers
119 views

find $O$ and $\Omega$ bounds as tight as possible for $T(n)=n+T(\frac n 2)+T(\frac n 4)+T(\frac n 8)+…+T(\frac n {2^k})$

find $O$ and $\Omega$ bounds as tight as possible for $T(n)=n+T(\frac n 2)+T(\frac n 4)+T(\frac n 8)+...+T(\frac n {2^k})$ while k is some constant and for any $n\leq3$ $\ T(n)=c$ for k=1 ...
0
votes
1answer
70 views

How do I go about manipulating this summation equation to solve it?

In my textbook, Introduction to Algorithms, the following is shown: And I believe I understand that. However, I have a similar equation to the one on the first line, but instead of ...
1
vote
2answers
71 views

How does my textbook solve this summation equation for the answer?

Summations have always been my weakness in mathematics, and it's showing here as I'm very confused how my textbook, Introduction to Algorithms, goes from basically the second half of the following ...
1
vote
1answer
54 views

How does my textbook come up with this statement? I don't believe it to be true.

My textbook (Introduction to Algorithms) states the following: When polynomially comparing $n^\epsilon$ and $lgn$, it states that $n^\epsilon$ is polynomially greater for any positive $\epsilon$. ...
1
vote
3answers
79 views

Where is the value of the variable $\epsilon$ obtained in the following explanation my professor gave?

My professor gave us this explanation from the textbook Introduction to Algorithms regarding the Master Method/Theorem: As a first example, consider $$T(n)=9T(n/3)+n.$$ For this recurrence, we ...
0
votes
0answers
69 views

How is the algorithm for recursively printing permutations of a set of numbers this equation our professor gave us?

I'm having a great amount of trouble understanding where my prof got $T(m, n) = n(T(m+1, n-1) + m+1 + n)$ if $n > 1$ as the recursive formula for the algorithm for recursively printing the ...
0
votes
1answer
69 views

Solving a recurrence with division

I'm having this recurrence that is giving me a lot of trouble. $$F(2^0) = 1$$ $$F(2^k) = \frac 1 2 F(2^{k-1}) + 2^k$$ I will edit my post since this does not seems to work... The initial ...
1
vote
3answers
51 views

How does my professor go from this logarithm to the following one of a different base?

I don't understand on the second last line how my professor goes from $2^{log_3(n)} = n^{log_3(2)}$ how is that relation formed?
1
vote
1answer
69 views

Is this recursion well-defined?

I have a recursion defined by $$ f(n)=\max\{0,-c+pf(n-1)+(1-p)f(n+1)\} $$ with $0.5<p \leq 1$ and $f(0)=R>0$ and $f(m)=0$ for some $m>0$. I am trying to show that $f(n)$ is decreasing in ...
2
votes
2answers
161 views

recurrence relation homework question

This is a homework question let $a_n$ number of n digit quaternary $(0,1,2,3)$ sequences in which there is never a$ 0 $anywhere to the right of a $3$. Solve for $a_n$ bot sure how to go about this. ...
2
votes
2answers
112 views

Purchasing Order: An Analysis on A ($\textrm{C++}$)-Approached Recurrence Relation

Suppose that we have $n$ dollars and that each day we buy either tape at a dollar, paper at a dollar, pens at two dollars, pencils at two dollars, or binders at three dollars. If $R_n$ is the number ...
1
vote
1answer
85 views

Orange Juice, Milk, or Beer

Suppose that we have $n$ dollars and that each day we buy either orange juice for a dollar, milk for two dollars, or beer for two dollars. If $R_n$ is the number of ways of spending all the money, ...
0
votes
3answers
168 views

(CHECK) $n$-bit Strings Containing a Pattern

$$\text{$\bf{PLEASE~~~CHECK~~~AUTHOR'S~~~ANSWER}$}$$ If $S_n$ denotes the number of $n$-bit strings that do not contain the pattern $00$, then what is the underlying recurrence relation and ...
3
votes
1answer
64 views

Recursively Defined Entities

So I am having some trouble understanding how one is to come up with the recursive definition to the following problem... We are given a rectangle of width $2$ and length $n$. Suppose we have ...
0
votes
1answer
224 views

$n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
1
vote
2answers
62 views

Recursion problem help

The following are the teachers example problems. The issue is that I don't understand the exact steps they took to go from $f(0)$ to $f(1)$ to $f(2)$ to $f(3)$. What I'm asking here is if someone ...
1
vote
3answers
61 views

Solve recurrence equation

Could you Show me how to solve this equation: $$x_n = \sqrt2x_{n-1} + \sqrt3$$ for $n \ge 1$ with $x_0 = 1$.
1
vote
2answers
35 views

Recursive formulae involving a linear operator

Given a basis $e_{1}$, $e_{2}$ in the plane, define the linear operator $F$ as $F(e_{1})=3e_{1}+e_{2}$ and $F(e_{2})=e_{2}$. Furthermore, define the sequence $u_{1},u_{2},\dots$ of vectors in the ...