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Type I -- Stellar Number


ibworking

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Hi!

I'm new to the site, mostly becaus up until now I have not had too many problems. However, I just hit a huge wall with this portfolio task.

I've been able to do the triangular numbers section, but was hit with a wall when attempting to find the formula for the 6-vertice stellar numbers in terms of n. I don't think it should be a recursive formula.

The formula we were shown to use in class was

Sn = n/2[(2U1)+(n-1)d]

, the formula for the sum of an arithmetic series. The first term is 1, and the difference between each term in the arithmetic series is 12, and those are the numbers I used, but it does not work for the formula. However, I was only off by a little bit. I followed the formula and worked it through correctly, so I don't know where I went wrong!

If anyone could offer suggestions of what I could look at to fix this, that would be great. I'm not looking for anyone to do the formula for me, but I am just wondering why it did not work and how I should approach the problem.

Thanks!

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You need to use quadratic sequences, finding the second difference between the numbers.

This page - http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/sequenceshirev1.shtml -

has quite a good explanation of the basics. Once you find the sequence for 6p, its quite easy to do the rest of the tasks.

As an extra tip all quadratic sequences have the formula: an2 + bn + c

I meant an sqaured, not an times 2.

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of course it doesnt work with this formula because its not arithmetic sequence, but in the other posts there was an explanation for this queation, it was saying that this sequence can be explained by quadratic formula. you should look at it, it was posted on May 2010.

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Ah, I think I found it!

There's a method beyond all of that that you can use that involves a sequence from the same portfolio. I won't say any more because that might be giving away answers to people still working on it.

Good luck to everyone else!

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Ok guys I found both the general terms for the triangular number sequence and the star shaped sequence. I am stumped on the question they ask after the general term. The question is "Now repeat the steps above for other values of p." I really do not understand what they are asking here. Can anyone help me out here? And if any of you guys need help in trying to get the general terms, I am more then willing to help you out. I ahve another favor to ask. Does anyone have a program or something that generates the triangular sequence images and the stellar shaped images, because these will be of good use in our internal assesments.If possible can anyone upload it? Or send it to my e-mail? Thanks

Edited by IB Killer Flamez
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Ok guys I found both the general terms for the triangular number sequence and the star shaped sequence. I am stumped on the question they ask after the general term. The question is "Now repeat the steps above for other values of p." I really do not understand what they are asking here. Can anyone help me out here? And if any of you guys need help in trying to get the general terms, I am more then willing to help you out. I ahve another favor to ask. Does anyone have a program or something that generates the triangular sequence images and the stellar shaped images, because these will be of good use in our internal assesments.If possible can anyone upload it? Or send it to my e-mail? Thanks

They mean to find the general statement for stars with p vertices. So you know a star is 360/p where 360 is divisible by p. So numbers like 4, 5, 6, 8. The tricky part is finding the general statement of a p-stellar number in terms of p and n. Fun!

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I found all the formulas in this portfolio however my teacher and I are concerned about use of technology in this portfolio topic because it seems that it's required to draw stars with geogebra. The problem with geogebra is that geogebra is hard to learn. Does anyone here to suggest any software for this portfolio topic? I can share the correct equations tomorrow because my portfolio draft is not with me.

Thanks a lot

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Hey everyone,

Since finishing my math portfolio, I figured I would post the images I used for everyone to have and use. Yay!

I have the images hosted here:

http://sites.google....larnumbers/home

Someone's been kind enough to share those ^^

& please don't give your answers to others if they ask. We're trying to help students help themselves be better at this IA business, but what's the point if it becomes a big swapping of answers?

=)

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I used calculus.

Derivatives and then integration to find the general statement. It lays out so nicely. The only problem comes when I try to prove the general statement in terms of p and n. I feel that it is too shaky. But that is what I did.

I'm quite sure you don't have to prove the general statement at SL... I think it says " provide a correct informal justification" in the mark scheme

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What defines a star anyway? The formula I have works for whenever p is equal to 2 or more, but can a 2 vertices figure be stellar? Won't it just look like a rhombus?

That's a good question. I actually have a similar question. If n represents numbers in a sequence, wouldn't it be limited to integers greater than 0? Using my equation, I get positive answers for Sn even when n is 0 or negative, but wouldn't such answers be impossible? Also, how many verticies are required to make an actual star shape? Heck, I've got values for a star with a single vertice. What would that even look like? According to my equation, Sn increases with n, even when p=1.

Help, please?

Edited by Heavy Swinger
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Hi guys :D

i am new to this site but i guess it is pretty helpful!!!

I got my Maths SL Posrtfolio about Stellar Numbers 2010 last Thursday and we did some in class already. The teachers are not allowed to actually help us so one of our students caltulated the first question which was: "The following diagrams show a triangular pattern of evenly spaced dots. The number of dots in each diagram are examples of triangular numbers(1, 3, 6,..).

Complete the triangular numbers sequences with three more terms"

I did all that drawing of the dots and stuff but the next question is : "Find a gerneral Statement that represents the nth triangular number in terms of n."

I tried my best on this which looks like thsi.

n 1 2 3 4 5 6 7 8

Sn 1 3 6 10 15 21 28 36

2 3 4 5 6 7 8

1 1 1 1 1 1

: 2

0.5

0.5n² 0.5 2 4.5 8 12.5 18 24.5 32

Sn-0.5n² 0.5 1 1.5 2 2.5 3 3.5 4

0.5 0.5 0.5 0.5 0.5 0.5 0.5

-0.5n -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4

And then we got this in clas: Sn= 0.5n²+0.5n but i am not sure if that should not be Sn= 0.5n²-0.5n so minus 0.5n

we did all this in class together but i am still not sure because i get a different answer which would be the -0.5n.

I hope anyone can help me with that:S

Thank you very much!

Kathy

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About the star... yeah, you stumbled upon a parameter/limitation. I think it needs more than 4 vertices, although I'm not the expert.

Kathy, I didn't go through finding the equation, but when you plug in numbers for n into your formula for Sn, the +.5n works and the -.5n doesn't. Do you know what I mean?

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About the star... yeah, you stumbled upon a parameter/limitation. I think it needs more than 4 vertices, although I'm not the expert.

Kathy, I didn't go through finding the equation, but when you plug in numbers for n into your formula for Sn, the +.5n works and the -.5n doesn't. Do you know what I mean?

Hi

yeah i think i know. I tried it out as well and it does :D do you think it is a good idea to test/prove the equation by substituting some number in to show it works? that might show that the equation is correct and i maybe should order the whole data into a table that might be more effective.

Thank you very much!!

kathy

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It's good to show that the equation works by putting numbers in, but this isn't the way to prove it since you can't possibly check all the numbers :D But yes, a table with several numbers shown to work wouldn't hurt.

What you need to do is go back and look at why it's +.5n and not -.5n.

I just solved for the coefficients, and I got what your classmates got.

an2 +bn +c= Sn

I plugged in n=1 and n=2 to find a and b [i assumed c = 0 because of what you had done earlier, but you'd need to plug in n=3 or any other number to find c].

I got a + b = 1 & 4a + 2b = 3

Do you follow?

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It's good to show that the equation works by putting numbers in, but this isn't the way to prove it since you can't possibly check all the numbers :D But yes, a table with several numbers shown to work wouldn't hurt.

What you need to do is go back and look at why it's +.5n and not -.5n.

I just solved for the coefficients, and I got what your classmates got.

an2 +bn +c= Sn

I plugged in n=1 and n=2 to find a and b [i assumed c = 0 because of what you had done earlier, but you'd need to plug in n=3 or any other number to find c].

I got a + b = 1 & 4a + 2b = 3

Do you follow?

yeah iu get the first bit with a+b=1 but why does 4a+2b=3? if i put 2=n into the equation i get a2²+2b=2 or? and that would then be 4a+2b=2 ??? Or is that wrong? but how do i prove that the equation is right?

kathy

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