Whats up guysssssssssss.
A.) 1: 1
2: 3
3: 6
4: 10
5: 15
6: 21
7: 28
8: 36
9: 45
B.) i'm not sure what the pattern is, but they are all divisable by the number added to them..
C.) If i had to pick a formula it would be something like n(n + n+1) or something like that. It has to be something saying taht there is a divisor in there somewhere..
D.) I do think there is a formula that can interpret the pattern of this. It would work because every number has certain qualities involving certain addition, subtraction, multiplication, and division rules.
Sunday, September 12, 2010
Saturday, September 11, 2010
Week 2 Blog Prompt
So for those of you who haven't seen (or have forgotten,) here is a repost of the prompt for this week:
Part a.) What is the sum of the first n integers for n=1, 2, 3, 4, 5, 6, 7, 8, 9, and 10?
*Note, that is to say, for n=6 you will simply calculate 1+2+3+4+5+6
Part b.) Is there a pattern as n gets larger? What do you intuitively think the pattern might be? If you don't see one immediately, try a few larger values of n (around 15 to 25.)
Part c.) Do you think it's possible to write a formula for the sum of the first n integers for any n (for instance, if I want to know the sum of the first 1,000,000 natural numbers, but don't want to write all of them out, is there a shorter formula that I can simply plug n=1,000,000 into?) If so, what do you think it might be (or if you're not sure, why do you think there is one?) and if not, why don't you think so?
Part d.) If you think there is a formula, can you justify (or prove) that it is correct? If you don't think there is one, can you prove that there isn't? (It's okay if you can't do this part, but just elaborate a bit.)
Keep in mind that you are not expected to do any research (and in fact, it looks worse if you do.) It will be relatively obvious if you use an outside source or another student's blog post as a basis for your own.
Part a.) What is the sum of the first n integers for n=1, 2, 3, 4, 5, 6, 7, 8, 9, and 10?
*Note, that is to say, for n=6 you will simply calculate 1+2+3+4+5+6
Part b.) Is there a pattern as n gets larger? What do you intuitively think the pattern might be? If you don't see one immediately, try a few larger values of n (around 15 to 25.)
Part c.) Do you think it's possible to write a formula for the sum of the first n integers for any n (for instance, if I want to know the sum of the first 1,000,000 natural numbers, but don't want to write all of them out, is there a shorter formula that I can simply plug n=1,000,000 into?) If so, what do you think it might be (or if you're not sure, why do you think there is one?) and if not, why don't you think so?
Part d.) If you think there is a formula, can you justify (or prove) that it is correct? If you don't think there is one, can you prove that there isn't? (It's okay if you can't do this part, but just elaborate a bit.)
Keep in mind that you are not expected to do any research (and in fact, it looks worse if you do.) It will be relatively obvious if you use an outside source or another student's blog post as a basis for your own.
Heather's #2
:D YESSSSSSSSSSS! I know how to do this because Alex taught me this for my sequences and series test at nationals!! Ha anywayyyy this is what you would get using this method > (*n=6 >> 1+2+3+4+5+6)
A.)
n=1: 1
n=2: 3
n=3: 6
n=4: 10
n=5: 15
n=6: 21
n=7: 28
n=8: 36
n=9: 45
n=10: 55
B.) Yes there is definitely a pattern. First if you look at the row of sums, you notice that you get the next number by adding the next "n"...As in if you take 1, your first sum, and add 2 you get your second sum which is 3. Adding 3 to your second sum you get 6, your third sum. Then adding 4 to 6 you get your 4th sum which is 10 and so on...So in other words (looking at part A) you would add the two red numbers to get the blue number. Also, looking more closely at the sums, I notice that each n is either a divisor of or has a divisor of the sum number. For instance, for n=7, 7 is a divisor of 28..meaning that 7 can go into 28 evenly. Also, for n=4, 2 is a divisor of both 4 and 10 since 4 cannot go into 10 evenly.
C./D.) Yes it is possible to have a formula for this type of problem...but seeing as how I learned it already, I'll explain why I think there is a formula for this. First of all, I see a reason for having a formula for this because we all know that no one wants to sit down and add up all the numbers up to 1,000,000 and surely no mathematician had the time for that either. I assumed there had to be a formula since there is a common pattern here. No matter how large the sum becomes, the pattern is still there. However, that isn't always the case, but I figured there must be a pattern here since the sums are in ascending order. Sooo with that, the formula I learned is n(n+1)/2. (I honestly did learn this, I didn't look it up) To use it, all you have to do is plug in n. So if I really do want to find the sum of the first 1,000,000 natural numbers then I would get 1,000,000(1,000,001)/2 which would simplify to 500000500000. I'd say this formula is pretty accurate for natural numbers/positive integers specifically; however I'm not sure if it would work the same say if you were asked to find the sum of the first 100 negative integers..
A.)
n=1: 1
n=2: 3
n=3: 6
n=4: 10
n=5: 15
n=6: 21
n=7: 28
n=8: 36
n=9: 45
n=10: 55
B.) Yes there is definitely a pattern. First if you look at the row of sums, you notice that you get the next number by adding the next "n"...As in if you take 1, your first sum, and add 2 you get your second sum which is 3. Adding 3 to your second sum you get 6, your third sum. Then adding 4 to 6 you get your 4th sum which is 10 and so on...So in other words (looking at part A) you would add the two red numbers to get the blue number. Also, looking more closely at the sums, I notice that each n is either a divisor of or has a divisor of the sum number. For instance, for n=7, 7 is a divisor of 28..meaning that 7 can go into 28 evenly. Also, for n=4, 2 is a divisor of both 4 and 10 since 4 cannot go into 10 evenly.
C./D.) Yes it is possible to have a formula for this type of problem...but seeing as how I learned it already, I'll explain why I think there is a formula for this. First of all, I see a reason for having a formula for this because we all know that no one wants to sit down and add up all the numbers up to 1,000,000 and surely no mathematician had the time for that either. I assumed there had to be a formula since there is a common pattern here. No matter how large the sum becomes, the pattern is still there. However, that isn't always the case, but I figured there must be a pattern here since the sums are in ascending order. Sooo with that, the formula I learned is n(n+1)/2. (I honestly did learn this, I didn't look it up) To use it, all you have to do is plug in n. So if I really do want to find the sum of the first 1,000,000 natural numbers then I would get 1,000,000(1,000,001)/2 which would simplify to 500000500000. I'd say this formula is pretty accurate for natural numbers/positive integers specifically; however I'm not sure if it would work the same say if you were asked to find the sum of the first 100 negative integers..
Tuesday, September 7, 2010
Ryan's #1
What is number theory?
When I hear number theory, I think of the branch of math that explains how numbers work. I think of the properties of how numbers and patterns work. I think it is about sequences, series, modular numbers, and other subjects dealing with how numbers work.
ryanbreauddawg(:)
When I hear number theory, I think of the branch of math that explains how numbers work. I think of the properties of how numbers and patterns work. I think it is about sequences, series, modular numbers, and other subjects dealing with how numbers work.
ryanbreauddawg(:)
Monday, September 6, 2010
Mal's Post 1
What is number theory?
Okay. Simple right? I believe number theory is a branch of mathematics that deals with just what the title implies...Numbers. Imagine that. In a way, it could be viewed as a higher level arithmetic considering much of the coursework involves studying numbers and manipulating them. Come to think of it, number theory is also the branch of mathematics that deals with the properties of integers, rational, natural etc. numbers...
According to the Mathematical Atlas (see link below), there are many sub-branches, if you will, of number theory. The way I look at it is the following: little kid number theory and big kid number theory. Remember when you were in 4th grade going over, for what probably was the thousandth time, prime and composite numbers? yeah, number theory. Now think back to the first/second week of school. What were we doing? Prime and composite numbers correct? Except, that sieve of eranth(whatever) was NOT something we used in 4th grade. Hence, the big kid title.
Kthnxbai...:DD
ohh...almost forgot. Source= this website: http://www.math.niu.edu/~rusin/known-math/index/11-XX.html
Okay. Simple right? I believe number theory is a branch of mathematics that deals with just what the title implies...Numbers. Imagine that. In a way, it could be viewed as a higher level arithmetic considering much of the coursework involves studying numbers and manipulating them. Come to think of it, number theory is also the branch of mathematics that deals with the properties of integers, rational, natural etc. numbers...
According to the Mathematical Atlas (see link below), there are many sub-branches, if you will, of number theory. The way I look at it is the following: little kid number theory and big kid number theory. Remember when you were in 4th grade going over, for what probably was the thousandth time, prime and composite numbers? yeah, number theory. Now think back to the first/second week of school. What were we doing? Prime and composite numbers correct? Except, that sieve of eranth(whatever) was NOT something we used in 4th grade. Hence, the big kid title.
Kthnxbai...:DD
ohh...almost forgot. Source= this website: http://www.math.niu.edu/~rusin/known-math/index/11-XX.html
Sunday, September 5, 2010
What is Number Theory???
Number Theory is a branch of mathematics that studies intergers and their relation and properties to each other. It deals with different classifications that intergers and numbers can be grouped into. It is basically an easier way to solving complicating math processes by using easier steps and easy to grasp methods
Blog 1
Wow, what is number theory?
Pshh, that's an easy question.
It's......damn, forgot........
I'm just gonna wing it right here, so don't mind me. :P
What do I think number theory is?
It's the theory of numbers, duh.
I do remember something about cracking codes though. That sounds really fun. *sarcasm*
I seriously have no idea....
*bursts into a rap*
Yo this is a story all about how numbers get theorized into oblivion.
They got codes being cracked, alarms being set off, and a whole buncha dogs chasing after us.
So then I heard ma dawgs talkin bout how this kind of theory can help us get away with this one.
They said "If we hit this jump at the right angle, we gonna get out of this joint."
I told em that I couldn't figure it out too easily.
They questioned me and then they told this to me:
"Boy you don't get it? Man go take this class. It's gonna help you out in all kinds of fields like that.
And if you actually do good, they gonna take you in the math club, which is mandatory in that class man."
I questioned em with such a surprised look and asked em "Dudes, how do yall even know about this class."
They didn't tell me and just sent me here.
Now here I am, in this class, not knowing a thing about Number Theory.
It'll be a while till I figure it out.
Pshh, that's an easy question.
It's......damn, forgot........
I'm just gonna wing it right here, so don't mind me. :P
What do I think number theory is?
It's the theory of numbers, duh.
I do remember something about cracking codes though. That sounds really fun. *sarcasm*
I seriously have no idea....
*bursts into a rap*
Yo this is a story all about how numbers get theorized into oblivion.
They got codes being cracked, alarms being set off, and a whole buncha dogs chasing after us.
So then I heard ma dawgs talkin bout how this kind of theory can help us get away with this one.
They said "If we hit this jump at the right angle, we gonna get out of this joint."
I told em that I couldn't figure it out too easily.
They questioned me and then they told this to me:
"Boy you don't get it? Man go take this class. It's gonna help you out in all kinds of fields like that.
And if you actually do good, they gonna take you in the math club, which is mandatory in that class man."
I questioned em with such a surprised look and asked em "Dudes, how do yall even know about this class."
They didn't tell me and just sent me here.
Now here I am, in this class, not knowing a thing about Number Theory.
It'll be a while till I figure it out.
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