a simple pattern.?!

[BMAD]

James was in his favorite Math class when his professor wrote on the board the following elementary pattern:

2, 4, 6, 8, ___. Asked to find the next value, James quickly raised his hand and exclaimed that the answer was 10. The professor, with a concerned and shocked expression on his face, asked James to settle down. When all of the other students looked equally puzzled, the professor explained to them (in an admonishing tone) that the correct number is in fact 34. Seeing the class was still perplexed, the teacher added even the next term after the fifth one showing:

2, 4, 6, 8, 34, 132.

Now, the group was tasked to not only find the next term but they also had to find the value that came before the one listed on the board.

What are these two numbers?

[witzar]

(n-3)^4 + 2(n-3)^3 - (n-3)^2 + 6

(Note: word 'obvious' is used as a joke.)

Edited April 1, 2013 by witzar

[BMAD]

before the FIRST one listed on the board.

[bonanova]

I am sensing, but not actually seeing, something Fibonacci-like.

[BMAD]

I am sensing, but not actually seeing, something Fibonacci-like.

but what does finonacci like mean?

[dark_magician_92]

(n-3)^4 + 2(n-3)^3 - (n-3)^2 + 6

(Note: word 'obvious' is used as a joke.)

How did you get that? Awesome!

[BMAD]

(n-3)^4 + 2(n-3)^3 - (n-3)^2 + 6

(Note: word 'obvious' is used as a joke.)

Unless I made a computation mistake, I am getting 137 when n=6 not 132 using your formula.

[karthik]

(n-3)^4 + 2(n-3)^3 - (n-3)^2 + 6

(Note: word 'obvious' is used as a joke.)

Unless I made a computation mistake, I am getting 137 when n=6 not 132 using your formula.

you did make a computation mistake, it's 132 when n=6.

Edited April 2, 2013 by karthik

[BMAD]

Is it too late to blame the calculator?

[BMAD]

When people see 'simple' patterns they generally go for the simplest means to explain the pattern and attempt to explain a pattern well beyond the given numbers when in reality this is an impossible to do. We are most guilty of this in k-12 schooling.

The pattern 2,4,6,8,... for example

Most people (I believe) would assume 2n (where n is ther term) like poor james did but there is no reason for this to be any more true then:

2n + (n-1)(n-2)(n-3)(n-4)... for however many 2n terms I want to show. We naturally, it seems, like to apply occam's razor when examining these problems.

Again, looking at another simple pattern like:

3, 4, 5, 6, 7,... most would say n+2 but again this isn't necessarilly the case as I can create infinitly many functions that generate this pattern:

n+2+(n-1)(n-2)(n-3)(n-4)(n-5)...

Therefore revealing the next term in these types of patterns DOES NOT divulge the true nature of the pattern unless it is drastically different or if and only if it is accepted that the pattern never deviates from its visually given pattern.

Edited April 2, 2013 by BMAD

[Prime]

We had a recent discussion of Lagrange Interpolating Polynomial here: http://brainden.com/forum/index.php/topic/15537-closed-form-expression-on-steroids/?p=327698