Thursday, May 29, 2008

Cubic Sequences

Today I have something similar to the cubic function discovery I did exactly two weeks ago. It's somewhat more complex, and therefore more fun. Long? Yes. Fun? Oh, yeah!

Find the nth term: 2, 8, 20, 40, 70, 112, 168, ...

Once again, the genre of "short question, super long answer". Those are the most fun, so let the fun begin.

When you get a sequence and you need to find the nth term, go for differences between numbers. Those differences hide a big secret within them. If the differences are the same, it's a simple arithmetic series. If the differences are not the same, do the differences of the differences. If now they're equal, the relation between the numbers goes through a quadratic expression. If the differences are still not the same, take the differences again and again until they are equal. The number of times you take the differences is the degree of the expression that connects the numbers.

If you see that the differences have a certain ratio to them (for example, try to find the differences for 4, 10, 28, 82, 244, 730), don't bother going more than once or twice. You are dealing with an exponential relation here.

So, let's find the differences of the sequence we're given:
2, 8, 20, 40, 70, 112, 168
6 12 20 30 42 56
6 8 10 12 14
2 2 2 2

Jackpot. We reached equal differences after three times, so the numbers are connected via a cubic expression. To find that expression, we need to set up a system of 4 equations with 4 variables (if you don't know or don't remember, open the link at the top of the post).

Let's call the points that this "function" goes through (1, 2) (2, 8) (3, 20) (4, 40). Now let's set up the system:
a + b + c + d = 2
8a + 4b + 2c + d = 8
27a + 9b + 3c + d = 20
64a + 16b + 4c + d = 40

Subtract the equations from each other to eliminate d:
7a + 3b + c = 6
19a + 5b + c = 12
37a + 7b + c = 20

Subtract them again to eliminate c:
12a + 2b = 6
18a + 2b = 8

And once again:
6a = 2
a = 1/3

Let's start plugging:
12/3 + 2b = 6
4 + 2b = 6
2b = 2
b = 1

Again:
7/3 + 3 + c = 6
c = 2/3

Finally:
1/3 + 1 + 2/3 + d = 2
d = 0

Well, that makes it easier. The nth term is given by:
n3/3 + n2 + 2n/3

Go ahead, plug your number for n and see that it works. For all terms, even the ones which were not included in the calculation.

Enjoy your new knowledge,
Nadav

nadavs

72 comments:

Anonymous said...

I have tried this technique but I still can't find the nth term for this sequence:
1, 4, 10, 20, 35...

Please help!

nadavs said...

The difference sequences are:
3, 6, 10, 15
3, 4, 5
2, 2, 2

So it's a cubic.

Which means:

a + b + c + d = 1
8a + 4b + 2c + d = 4
27a + 9b + 3c + d = 10
64a + 16b + 4c + d = 20

Subtraction:
7a + 3b + c = 3
19a + 5b + c = 6
37a + 7b + c = 10

More subtraction:
12a + 2b = 3
18a + 2b = 4

Final subtraction:
6a = 1

a = 1/6

Plug in for "a" and continue from there:
2 + 2b = 3
b = 1/2

7/6 + 3/2 + c = 3
7/6 + 9/6 + c = 18/6
c = 1/3

1/6 + 1/2 + 1/3 + d = 1
d = 0

So the nth term is:
n^3 / 6 + n^2 / 2 + n / 3

Try it, it works.

nadavs

Anonymous said...

I does work!
Thank you so much.
Can that be simplified any further

Anonymous said...

I tried this for the sequence
2, 5, 9, 15, 24
I got 1/6n^3-1/2n^2+3.5n-2 but it's wrong. (HELP)

nadavs said...

Let's see:

2, 5, 9, 15, 24

Differences:

3, 4, 6, 9
1, 2, 3
1, 1

So it's a cubic.

Now let's place the numbers:

a + b + c + d = 2
8a + 4b + 2c + d = 5
27a + 9b + 3c + d = 9
64a + 16b + 4c + d = 15

Differences:

7a + 3b + c = 3
19a + 5b + c = 4
37a + 7b + c = 6

More differences:

12a + 2b = 1
18a + 2b = 2
6a = 1
a = 1/6

Plug in numbers we find:

12/6 + 2b = 1
2 + 2b = 1
2b = -1
b = -1/2

7/6 - 3/2 + c = 3
7/6 - 9/6 + c = 18/6
-2/6 + c = 18/6
c = 20/6 = 10/3

1/6 - 1/2 + 10/3 + d = 2
1/6 - 3/6 + 20/6 + d = 12/6
18/6 + d = 12/6
d = -6/6 = -1

So the cubic is:

(1/6)n^3 - (1/2)n^2 + (10/3)n - 1

Hope that helps,
nadavs

Anonymous said...

Im having a problem with the following sequence:
0,6,48,204,624,1554,3360
please help find any relationship with the pascals triangle or whether it has a purpose.
thanks

Anonymous said...

ohmygod thanks so much! this helped a lot!

Anonymous said...

Hello, can you help me solve for the nth term when n^4?
Here...
1;5;15;35;70;...

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