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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

I know that I'm capable, it's just that I'm 24 almost 25, and I haven't taken any math since I was 16 and even then in High School I never applied myself. It's a shame really, because now I'm in college and I'm taking college Algebra, which is almost sad really. I just feel like I'm really far behind, and I'm really not sure how to go about catching up quickly.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Okay, we will talk about that later.

For now the third idea:

{1,5,14,30,55,91,140,204,285,385}

{4,9,16,25,36,49,64,81,100}

{5,7,9,11,13,15,17,19}

{2,2,2,2,2,2,2}

when you come to a constant difference in this case 2 you stop. Because there are 4 rows that means the answer is third degree polynomial.

There is now a simple formula for your recurrence.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

Now that's interesting, when I was trying to work out a formula that didn't involve a recurrence, I actually worked out these first two patterns and were trying to figure a way to use them both, but ultimately couldn't.

{1,5,14,30,55,91,140,204,285,385}

{4,9,16,25,36,49,64,81,100}

And actually, I don't remember what I had done, but I had come to a pattern that went up by 4 constantly. I was working on the computer though and didn't save my notes.

*Last edited by therussequilibrium (2012-12-21 21:15:03)*

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi therussequilibrium

How to get a formula?

There are loads of methods. Here's one using your celcius to fahrenheit example.

There are two key points on the celcius scale: freezing point of water ... 0 and boiling point of water ... 100

The equivalent values in farhenheit are 32 and 212.

So on a graph, with C across and F up, (0,32) and (100,212) are known points for the conversion. (see picture ... this is a sketch only ... it is not accurately drawn)

So for 100 across you have to go 180 up to stay on the line. ie. gradient = 180/100 = 9/5

So if the equation is

Substitute C = 0 and F = 32 gives

Your empirical formula was very close so you did well!

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi Bob;

How are you?

Hi therussequilibrium;

{1,5,14,30,55,91,140,204,285,385}

{4,9,16,25,36,49,64,81,100}

{5,7,9,11,13,15,17,19}

{2,2,2,2,2,2,2}

this is called a difference table. Each element in the row below is made by subtracting two elements in the row above.

5 - 1 = 4, 140 - 91 = 49.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi bobbym,

I'm fine thanks. Throat almost back to normal. How are you ?

I think I've got another approach to the 'Pyramid numbers' example. But I need to have a long think first .....

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

I actually do this with patterns already, naturally, I just had no idea how to apply those patterns, or if they could even really be applied to do anything.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi Bob;

Glad you are back to almost 100%. There are many approaches to his problem. We are just looking one, the recurrence he came up with.

Hi therussequilibrium;

You can use the numbers you have to come up with this formula in terms of binomials.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

I'm not sure I know what you mean in your last post.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

Have you had combinations yet?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

No, I haven't learned of Combinations yet. /sigh so much to learn lol

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Okay, we do not need them. They just simplify the notation. Do you understand the difference table?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

Yes, I completely understand the table.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Let's work on that for a second. When you had 0 squares how many squares did you count?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

0

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Very good so your sequence really looks like this

{0,1,5,14,30,55,91,140,204,285,385...}

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

Correct, and the second sequence really starts at 1 and not at 4, etc.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Yes, the difference table now looks like this.

Take the first number in every row and say

a = 0;

b = 1

c = 3

d = 2

the rest are zero. Take those and plug into this formula.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

Not sure how to type out a fraction on the forum, but I think this is the answer:

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

That is not correct. Let me show you what I meant.

I dropped the ellipses on the end which are just there to show that there might be more terms.

When you substitute a = 0, b = 1, c = 3, d = 2 into that you get,

you can leave it like that and say

or you can simplify it,

That is what we got by the other methods. So this method solves your recurrence by just making a table and plugging into a formula.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

hmm, that's exactly the equation I entered and that's the answer it gave me; and I'm not sure how to simplify a problem with radicals in it, on my own.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

I am sorry, that is a factorial.

2! means 2 * 1, it equals 2

3! means 3 * 2 * 1, it equals 6

I did away with the factorials.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

Well, I can't figure out how to simplify this problem. But question, where does this equation that we are using come from?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

Offhand I do not remember the derivation of that. It is from the difference calculus. But it does work.

Try simplifying it by hand. If you want to I will help but it is really not necessary.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**therussequilibrium****Member**- Registered: 2012-12-21
- Posts: 36

Yeah, I'm trying by hand, I'll get it eventually. You've given me enough help, but thank you. I do appreciate you taking your time to walk through this with me.

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