1
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2answers
23 views

Discrete math and recursion problem.

I was recently reading up examples on recursion and how it relates to induction and there's this question I am not sure about. Q: Let $$b_1=3$$ $$b_n=n(n+2)$$ From that question I wanted to do the ...
0
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1answer
21 views

Check these recursive definitions for me?

Looking for Give a recursive definition of A) the set of odd positive integers B) the set of positive integer powers of 3 C) the set of polynomials with integer coefficients I have a. Basis: ...
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2answers
46 views

Define this recursively : $f(n) = 3n - 4$

Define this recursively : $f(n) = 3n - 4$ I thought this is how you recursively define a function? $f^{-1}(n): y = 3n-4$ $y+4 = 3n$ $f^{-1}(n) = (y+4/3)$ But the answer is $f(n) = f(n-1) + B$ ...
1
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2answers
78 views

Use the recursive definition of summation together with mathematical induction to prove a sequence

Use the recursive definition of summation together with mathematical induction to prove that for all positive integers $n$ if $a_1, a_2,\ldots, a_n$ are real numbers, then $$\sum_{k=1}^n(3a_k - 2k + ...
1
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2answers
89 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 ...
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1answer
38 views

Find a recursive algorithm to find $a^{2^n}$

Edit1: Used Latex. =] Edit2: Thanks for the guidance to the users below. Really helped me out editing the post and guidance on the math problem. The question gave me a hint: $a^{2^{n+1}} = (a^{2^n}) ...
1
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1answer
29 views

How to find a pattern in this recursive sequence algorithmically?

I'm trying to find the closed-form of a sequence algorithmically. Here is the recursive sequence: $$w_k=w_{k-2}+k, \forall k \in \Bbb{Z} | k \geq 3, w_1=1, w_2=2$$ which produces this sequence: ...
2
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2answers
42 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 ...
3
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4answers
45 views

Is this a correct recursive sequence definition?

Take this definition: Is this definition of $s_k$ for $k\ge2$ correct? $s_k=6a_{k-1}-5a_{k-2}$ but where does the $a$ term come from? The book swaps $a$ and $s$ interchangeably.
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2answers
47 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 ...
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1answer
42 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 ...
1
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2answers
23 views

Question over recursive definitions

Let $E$ be the set of even integers. Then the Base Case: $0\in E$ And the constructor case would be If $n \in E$ then so are $n+2$ and $-n$. This makes sense. But would the Base case and constructor ...
2
votes
5answers
89 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 ...
0
votes
1answer
40 views

Discrete Math Recursive Functions for Strings

Recursive Functions for Strings Construct a recursive definition for the following string function over the alphabet {x,y}: f(x) returns the string where every x is replaced by xy and every y is ...
0
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2answers
53 views

Prove via induction this recursively defined sequence

Let $P(n) = 2P(n-1) + n, P(1) = 3.$ Use induction to show that $$P(n) = 3(2^n) - n - 2$$ Highly verbose solutions are greatly appreciated.
2
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1answer
184 views

How to solve this recurrence relation with Sigma notation (f(n, m) = f(n - 1, m) + f(n, m- 1) + c?

This recurrence relation was inferred from the function $f(n, m) = f(n - 1, m) + f(n, m-1) + c$. After expanding the latter, I ended up with the following: $$f(n,m)=\begin{cases} 0,&\text{if ...
0
votes
1answer
23 views

How do you use induction on a recursive sequence using different variables?

I've been working on some recursive sequences for my Discrete class. I've understood most of them, but I've come to a new question which I'm confused about. A sequence $C_{0}$, $C_{1}$, $C_{2}$ is ...
0
votes
3answers
35 views

Figuring out the steps in a Recursive Function

I have the following recursive function: $f(0) = 7$ $f(n+1) = f(n) + 6n + 1$ for all integers $n => 0 $ I know the answer is $f(n) = 3n^2 + 2n + 7$ I would like to know the steps to get to this ...
0
votes
1answer
48 views

Mathematical Induction Recursion

Consider the recursion given by $f(n) = 2f(n−1)− f(n−2)+6$ for $n ≥ 2$ with $f (0) = 2$ and $f (1) = 4.$ Use mathematical induction to prove that $f (n) = 3n^2 −n+2$ for all integers $n ≥ 0.$
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2answers
163 views

Proving a recurrence relation for strings of characters containing an even number of $a$'s

We consider strings of $n$ characters, each character being $a$, $b$, $c$, or $d$, that contain an even number of $a$'s. (Recall that $0$ is even.) Let $E_n$ be the number of such strings. ...
1
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1answer
61 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 ...
2
votes
3answers
49 views

Recursion definition

Give a recursive defintion of the following set: $\{ 5^m 7^n \mid m, n \in N \}$ I don't have the slightest idea how to approach this question, id be really grateful if someone could provide me with ...
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votes
3answers
151 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 ...
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2answers
48 views

Discrete Mathematics - Recursion

Given the following question by my professor: Recursively define the set of natural numbers divisible by 3. My answer: ...
1
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1answer
81 views

Number of ways to arrange different poker chips. (Recursion)

Assume there are poker chips in four different colors and one of the colors is blue. In how many ways can n amount of chips be piled on top of each other without two blue ones being next to one ...
0
votes
1answer
49 views

factoring cubic polynomail equation using Krammer's rule.

1) I have question about factoring cubic polynomials. In my note it says "Any polynomial equation with positive powers whose coefficients add to 0 will have a root of 1. Another, if sum of the ...
0
votes
1answer
89 views

Recursively defines function

Find $f(2),f(3),f(4)$, and $f(5)$ if $f$ is defined recursively by $f(0)=-1$, $f(1)=2$, and for $n=1,2,\ldots$ $$f(n+1)= 3f(n)^2 - 4f(n-1)^2$$
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votes
1answer
46 views

Recursive Formula

I have found a question in my book and I have solve it, but I am not sure about the answer. The question goes like this: Find a recursive formula that will give the number of ways to order $N$ ice ...
0
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0answers
69 views

Golf Player Balls

A golf player has $K$ balls of the same type and $N$ balls of different types. I. In how many ways can we paint the balls such that each ball will be painted only once (the order of balls in the bag ...
1
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1answer
75 views

Basic Recursion Exercise

Find a recursive formula to divide N people in to couples and singles. The answer: $$A_n = A_{n-1} + A_{n - 2}(n - 1)$$ Some one can explain the answer , why do we need to multiply by n-1?
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2answers
107 views

Basic Discrete Mathematics Question

I was preparing my self before an exam and I found this question: For each of the following equations, find a positive integer $n$ that satisfies the equation. The notation $p(n,r)$ stands for ...
1
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2answers
469 views

Can't understand a recursive definition of concatenation of two strings

I'm reading Rosen's Discrete Mathematics and its applications(6ed), but I can't understand a recursive definition about concatenation of two strings: Two strings can be combined via the operation ...
1
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1answer
56 views

Need help with proving the recursion

Let $p_{k}(n)$ indicate the number of partitions of n into k parts. Prove: $$p_{k}(n) = p_{k-1}(n-1) + p_{k}(n-k)$$ Example: There are two partitions of $5$ into three parts. $5 = 3+1+1$ $5 = ...
2
votes
2answers
158 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. ...
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1answer
49 views

Bit strings, $\omega$, $\lambda$ - need help interpreting and describing

Problem For these recursive definitions of sets of bit strings, show 5 elements from each set, and identify what set it is (in a few words). Attempt 1 Basis: $1 \in S_!$ Recursion: $\omega \in ...
3
votes
1answer
63 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 ...
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3answers
82 views

Recursion Question - Trying to understand the concept

Just trying to grasp this concept and was hoping someone could help me a bit. I am taking a discrete math class. Can someone please explain this equation to me a bit? $f(0) = 3$ $f(n+1) = 2f(n) + ...
1
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2answers
61 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 ...
2
votes
2answers
51 views

Using polynomials as recursions

I made this observation in my discrete math course a while back. I explored it further online, so not all the ideas contained are mine alone. I still am confused about some things, though. Consider ...
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1answer
121 views

Recursive Definitions with Converse

I think I know how to solve i. and ii., but not iii: Base Case: $(0,0) \in S$ Recursive Step: If $(a,b)\in S$, then $(a+1,b+2)\in S$ and $(a+2, b+1)\in S$. (For i and ii): Prove that if $(a,b) \in ...
0
votes
2answers
285 views

Give a recursive definition with initial condition…how did they get the answer?

The function $f (n) = 5n + 2, n = 1, 2, 3, \ldots $ Im sure its a simple problem, but im really confused...how did they get the answer ? could someone explain $f (n) = f (n - 1) + 5, f (1) = 7$ ...
3
votes
3answers
85 views

Explanation of the recursion for number of surjections

I have a question about the recursion of the number of surjections from $\{1,\ldots,n\}$ to $\{1,\ldots,k\}$: $$\mathrm{Sur}(n,k) = k \cdot \mathrm{Sur}(n-1,k-1) + k \cdot \mathrm{Sur}(n-1,k).$$ My ...
5
votes
4answers
3k views

What is the solution to the following recurrence relation with square root?

This looks like a question asked earlier, but it isn't T(n) = T (sqrt(n)) + 1 ... if n>1 =1... if n=1 My professor gave this to me in class yesterday. This is where I'm stuck.. T(n) ...
1
vote
0answers
134 views

1s surpassing 0s in binary strings of odd length

Let $A(k)$ be the number of distinct binary strings of length $2k+1,$ for which the number of $1$s surpasses the number of $0$s for the first time at digit number $2k +1$, i.e., in the final digit in ...