# Find the next term

1. Mar 12, 2014

### johnqwertyful

"Find the next term"

After studying more math, I've been conflicted about this type of assignment commonly given to high schoolers. In the sequence 1, 4, 9, 16, ____ find the next term.

On one hand, it's a good exercise and teaches pattern recognition. On the other though, it seems to go against a lot of math. 25 is the "obvious" choice, but is 25 any more correct than say, -43? Just because n^2 is A rule, and probably the "simplest" doesn't make it any more correct than any other rule. It seems to teach students to draw conclusions from incomplete information.

It DOES seem useful to teach kids pattern recognition and intuition. But is it really worth it? Or would there be a better way?

2. Mar 12, 2014

### faiza hassan

It teaches students intuition
It has its own worth

Remember it is very important in real life to draw conclusions from incomplete information

3. Mar 12, 2014

### faiza hassan

Remember the difference between deduction and induction. Both are very important in mathematical reasoning

4. Mar 12, 2014

### arildno

Actually, I rather agree with you.
Getting 25 rests on an UNSTATED premise of Occam's Razor application, i.e, we are to find the arguably simplest pattern those 4 instances would be the first four examples of.

Furthermore, to discipline the mind to "think simple" when doing maths is, actually, precisely the challenge those not doing well in maths have difficulties in.

5. Mar 12, 2014

### SteamKing

Staff Emeritus
And if the next term in the sequence were -43, how to explain the sequence?

6. Mar 12, 2014

### DrGreg

I've no idea what was in johnqwertyful's mind but one possible answer is$$\frac{(n-2)(n-3)(n-4)(n-5)}{(1-2)(1-3)(1-4)(1-5)} + 4\frac{(n-1)(n-3)(n-4)(n-5)}{(2-1)(2-3)(2-4)(2-5)} + 9\frac{(n-1)(n-2)(n-4)(n-5)}{(3-1)(3-2)(3-4)(3-5)} \\ + 16\frac{(n-1)(n-2)(n-3)(n-5)}{(4-1)(4-2)(4-3)(4-5)} - 43\frac{(n-1)(n-2)(n-3)(n-4)}{(5-1)(5-2)(5-3)(5-4)}$$ Clearly an absurdly complicated answer but nevertheless valid. (And you could use this method to get any answer for any sequence.)

7. Mar 12, 2014

### micromass

Staff Emeritus
A sequence is just a function $\mathbb{N}\rightarrow \mathbb{R}$. Nobody said we can 'explain' (= give a general term for) any sequence. In fact, we can't for most sequences.

8. Mar 12, 2014

### lendav_rott

The rule of thumb is to keep it simple. There is no need to complicate things at all if it's not really necessary. Something I noticed while in uni, those in engineering insist on the simplest Working solution - the evolution of technology really does lie in simplicity, but it's so elusive, which makes it ingenious and that's why people go haywire.."zommmg he's a genius!!". Those in theoretical sciences, en masse, have some kind of inexplicable urge to blow everything out of proportion. :/ One of my friends in psychology carried out a small research between the students of 4 universities and pretty much confirmed my hunch. Of, course it isn't the whole information, but how much are you going to push the envelope? Eventually, what even is complete information? Perfectionists aren't welcome in my house, no offense. I'll get you some coffee or tea, but no cake, since that would become too much of a science.

9. Mar 12, 2014

### arildno

Where did the problem state "you MUST use Occam's razor here!"???

10. Mar 12, 2014

### arildno

An where, in these problem texts is the rule to apply Occam's Razor stated??

11. Mar 12, 2014

### johnqwertyful

Even better than that:
S1=1, S2=4, S3=9, S4=16, S5=-43

This is a perfectly valid sequence. You don't need to come up with a cute formula, you just need to ASK for what you want.

12. Mar 12, 2014

### micromass

Staff Emeritus

13. Mar 19, 2014

:rofl: