induction, recursive definition, number of MAC addresses, chapter summary, etc

[dattranvan22]

I have some a lesson exercise. I don't know how do that. Can you help me solve it?

1. Prove that 2015^0 + 2015^1 + 2015^2 + ... + 2015^n = (2015^(n +1) −1)/2014 by using induction.

2. Prove that 2^2n − 1 is divisible by 3 for all integers n ≥ 0 by using induction.

3. Give a recursive definition of the sequence a n, n = 1, 2, 3, ... if a n = 6n − 1.

4. MAC address of network card is represented by Hexadecimal. If they use 12 digits to assign MAC address, how many network card can be assigned using Hexadecimal without duplication?

6. A small company has 13 departments of total 190 employees. Each department has maximum 15 employees. Each month, each department organize a birthday party for all employees in the department that has birthday in that month. Prove that there is at least 1 month that has at least 2 birthday parties by 2 departments.

7. Show steps by steps of how to generate the next permutation of 231456?

8. Roll a fair die 10 times. X is assigned to the number appears, Y is assigned to 0 if number is odd and to the number of it is even. Calculate E(X), E(Y), V(X), V(Y).

9. Choose any topic introduced from chapter 5 to 12 and study the relation of that topic in discrete maths with computer science or information technology. Summary into 1 or 2 paragraphs with at least 15 sentences.

 

[Deleted member 4993]

I have some a lesson exercise. I don't know how do that. Can you help me solve it?
1. Prove that 2015^0 + 2015^1 + 2015^2 + ... + 2015^n = (2015^(n +1) −1)/
2014 by using induction.
2. Prove that 2^2n − 1 is divisible by 3 for all integers n ≥ 0 by using induction.
3. Give a recursive definition of the sequence a n, n = 1, 2, 3, ... if a n = 6n − 1.
4. MAC address of network card is represented by Hexadecimal. If they use 12 digits to
assign MAC address, how many network card can be assigned using Hexadecimal without
duplication?
6. A small company has 13 departments of total 190 employees. Each department has
maximum 15 employees. Each month, each department organize a birthday party for all
employees in the department that has birthday in that month. Prove that there is at least
1 month that has at least 2 birthday parties by 2 departments.
7. Show steps by steps of how to generate the next permutation of 231456?
8. Roll a fair die 10 times. X is assigned to the number appears, Y is assigned to 0 if number
is odd and to the number of it is even. Calculate E(X), E(Y), V(X), V(Y).
9. Choose any topic introduced from chapter 5 to 12 and study the relation of that topic
in discrete maths with computer science or information technology. Summary into 1 or 2
paragraphs with at least 15 sentences.

Looks like a Take-home exam!!! Are you supposed to ask for external help?!

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

 

[stapel]

1. Prove that 2015^0 + 2015^1 + 2015^2 + ... + 2015^n = (2015^(n +1) −1)/2014 by using induction.

2. Prove that 2^2n − 1 is divisible by 3 for all integers n ≥ 0 by using induction.

Induction is a fairly well-defined process. (here) How far have you gotten in working through the steps?

3. Give a recursive definition of the sequence a n, n = 1, 2, 3, ... if a n = 6n − 1.

I will guess that "a n" means "a_n" or (which is the same thing) "a_n". What have you found as being the pattern in the terms of this sequence? What changes, as one goes from one term to the next? How then can one define the next term in relation to the previous term?

4. MAC address of network card is represented by Hexadecimal. If they use 12 digits to assign MAC address, how many network card can be assigned using Hexadecimal without duplication?

This is a simple application of the "counting" or "multiplication principle". (The name varies; your book likely used one or the other of these two names.) How many "slots" do you have? How many options are available for the first slot? Then how many options are left for the next slot? And so forth. (You may be expected to express this in factorial form, rather than doing the complete multiplication.)

6. A small company has 13 departments of total 190 employees. Each department has maximum 15 employees. Each month, each department organize a birthday party for all employees in the department that has birthday in that month. Prove that there is at least 1 month that has at least 2 birthday parties by 2 departments.

How far have you gotten in comparing and relating the number of employees, the numbers of departments, and the numbers of months in a year?

7. Show steps by steps of how to generate the next permutation of 231456?

How does your book "order" permutations, that there would be one fixed "next" permutation? How far have you gotten in following the book's steps or methods?

8. Roll a fair die 10 times. X is assigned to the number appears, Y is assigned to 0 if number is odd and to the number of it is even. Calculate E(X), E(Y), V(X), V(Y).

What did you get when you rolled the die? How far have you gotten in filling out your table and applying the appropriate definitions?

9. Choose any topic introduced from chapter 5 to 12 and study the relation of that topic in discrete maths with computer science or information technology. Summary into 1 or 2 paragraphs with at least 15 sentences.

It's your choice of chapters in your book, and then your thoughts in relation to what you've read. There is no way for anybody to help you with this. Sorry. :shock:

 

[dattranvan22]

Looks like a Take-home exam!!! Are you supposed to ask for external help?!

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

No, No. This is exercises of object. I do it to get a document. I will use document for exam review.