What is this series and what are its next numbers?

NakulG · 2015-01-01 21:13:04

I came across this set of numbers,

15, 35, 53, 100, 243, ..., ...

I am trying to find out what is the next term in the series, can you help.
Thanks
Nakul

anonimnystefy · 2015-01-02 01:05:04

Where did you come across the? It might help us know how to approach it.

NakulG · 2015-01-02 15:18:14

i found the series on a site on the internet, i will try to send its link.

But on solving it (in excel on charts) the closest i have been able to get is the following equation

y = 1.5x4 - 9.8333x3 + 20.5x2 + 4.8333x - 2
but it is still not completely accurate (beyond 4 decimal points)

the next number in the series as per the above equation is
15, 35,53,100,243, 585, 1265

bobbym · 2015-01-02 15:25:00

Try this, it is an interpolation fit.

NakulG · 2015-01-02 15:25:07

can you help me find the series accurately...
thanks

bobbym · 2015-01-02 15:25:26

but it is still not completely accurate (beyond 4 decimal points)

Excel like most math programs works in floating point arithmetic. This can be a tricky beast for a least squares fit, which is what he did. To get exact results you must work with exact numbers, see post #4.

anonimnystefy · 2015-01-02 16:38:24

Your polynomial is the same as his.

NakulG · 2015-01-02 17:43:04

i ran the formula, the series i am getting is as follows:
15, 35, 53, 99.999..., 243, 585, 1265, 2458, 4375, 7263
The original series is:
15,35,53,100,243
The series as per floating point method -
15.0000, 35.0002, 53.0008, 100.0020, 243.0040, 585.0070, 1265.0112, 2458.0168, 4375.0240, 7263.0330

Your method is very accurate, only the 100 digit is not fully accurate can you help.
Can you also explain how you got to this that would be really interesting.

Thanks

NakulG · 2015-01-02 17:54:06

actually both the polymonials are same...but how did you get at the expression?

bobbym · 2015-01-02 19:30:27

Your polynomial is the same as his.

That is not correct, they are not the same.

NakulG · 2015-01-02 20:34:49

how are they different? am i missing something can you please explain.
thanks

bobbym · 2015-01-02 20:53:35

4.8333 is not the same as 29 / 6. One is a truncated number while the other is an exact rational.

anonimnystefy · 2015-01-02 20:54:24

bobbym wrote:

Your polynomial is the same as his.
That is not correct, they are not the same.

Okay, that is true.

NakulG · 2015-01-02 21:02:50

Ok, so how did you arrive at the exact fractions Form of the equation, that would be interesting to learn.
Thanks
Nakul

NakulG · 2015-01-02 21:04:20

Also the fourth term of the series, how do we exactly get that to 100?
Thanks

bobbym · 2015-01-02 21:09:51

To get the exact answer you will need to work with a simultaneous set of linear equations. You will also need to work in exact arithmetic, not floating point. This will require a computer.

There is also the method of differences which can be done by hand.

NakulG · 2015-01-02 23:33:48

Thanks so much.
This is helpful.
Also any place from where I can know about the method of differences.

Bob · 2015-01-03 02:18:57

hi NakulG

I think there are two things you want to know:

(1) How to obtain the quartic (n^4) equation. You can do this as follows:

Put in the values for n=1, 2, 3, 4, 5 and get 5 equations.

Go to an equation solving website such as http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi

Click the "Solving a system of linear equations" link.

This will give you the values of a, b, c, d, e.

(2) The method of differences. I tried to find a website for this, without success. So here's my own explanation. If you want more details post again.

Suppose you have a series that you think may be generated by a cubic, say

The next term would be

If you subtract each term from the one that comes next, the an^3 terms cancel out and you are left with a quadratic.

If you subtract each term from the one that comes next, the x^2 terms cancel and you get a linear equation.

Subtract again and every term is a constant.

So if this difference process ends up with a constant you know the original was a cubic.

eg. Suppose I have the sequence 17, 49, 121, 251, 457, 757, 1169, ...

Here is the table of differences:

The constant 18 at the third differences tells me this is generated by a cubic. What's more I can get the cubic term with this formula

a= the constant divided by (factorial 3) = 18/(3x2x1)= 18/6 = 3

So the cubic starts 3n^3 + .... a quadratic.

If you calculate the cubic terms and subtract them from the sequence, you'll get a sequence that has a quadratic formula. So you can do the method of differences again to get the coefficient for n^2. And so on, until you have the complete formula.

Hope that helps,

Bob

bobbym · 2015-01-03 04:28:04

I tried to find a website for this, without success

That is to be expected. Differences have all but disappeared. In the old days, Newton, Stirling, Gauss and Euler were masters of them and used them for everything. Nowadays, unless you are nearing 100 they will be unfamiliar.

We use the method of the nought sock one of my favorites because it does not look like a polynomial but it is!

Start by forming the difference table:

Now notice the first entry in each row,15, 20, -2, 31, 36

To form the interpolating polynomial we just use successive binomials like this:

There is only one little snag, can you see it?

NakulG · 2015-01-14 17:09:13

I formed the difference table. But in this table the fourth difference is 36 and it is not a constant like you showed in the earlier example. Interesting, but don't know why.

NakulG · 2015-01-14 18:30:01

Here we assume that it is a 4th degree polynomial. If it were a 7th degree polynomial then how do we calculate?

NakulG · 2015-01-14 18:48:18

Did I get the snag?

Last edited by NakulG (2015-01-14 18:56:51)

Olinguito · 2015-01-15 05:21:35

If you don’t need to simplify your polynomial formula, you can write it down immediately.

NakulG wrote:

15, 35, 53, 100, 243, ...

Answer:

[list=*]
[*]

[/*]
[/list]

Can you see the trick? If you substitute n=1, all but the first term disappear; if you substitute n=2, all but the second term disappear, etc.

Last edited by Olinguito (2015-01-15 23:55:28)

Math Is Fun Forum

#1 2015-01-01 21:13:04

What is this series and what are its next numbers?

#2 2015-01-02 01:05:04

Re: What is this series and what are its next numbers?

#3 2015-01-02 15:18:14

Re: What is this series and what are its next numbers?

#4 2015-01-02 15:25:00

Re: What is this series and what are its next numbers?

#5 2015-01-02 15:25:07

Re: What is this series and what are its next numbers?

#6 2015-01-02 15:25:26

Re: What is this series and what are its next numbers?

#7 2015-01-02 16:38:24

Re: What is this series and what are its next numbers?

#8 2015-01-02 17:43:04

Re: What is this series and what are its next numbers?

#9 2015-01-02 17:54:06

Re: What is this series and what are its next numbers?

#10 2015-01-02 19:30:27

Re: What is this series and what are its next numbers?

#11 2015-01-02 20:34:49

Re: What is this series and what are its next numbers?

#12 2015-01-02 20:53:35

Re: What is this series and what are its next numbers?

#13 2015-01-02 20:54:24

Re: What is this series and what are its next numbers?

#14 2015-01-02 21:02:50

Re: What is this series and what are its next numbers?

#15 2015-01-02 21:04:20

Re: What is this series and what are its next numbers?

#16 2015-01-02 21:09:51

Re: What is this series and what are its next numbers?

#17 2015-01-02 23:33:48

Re: What is this series and what are its next numbers?

#18 2015-01-03 02:18:57

Re: What is this series and what are its next numbers?

#19 2015-01-03 04:28:04

Re: What is this series and what are its next numbers?

#20 2015-01-14 17:09:13

Re: What is this series and what are its next numbers?

#21 2015-01-14 18:30:01

Re: What is this series and what are its next numbers?

#22 2015-01-14 18:48:18

Re: What is this series and what are its next numbers?

#23 2015-01-15 05:21:35

Re: What is this series and what are its next numbers?

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