Find the next term

"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?

Answers and Replies

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

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

arildno
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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?

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.

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

DrGreg
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And if the next term in the sequence were -43, how to explain the sequence?
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.)

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

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.

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.

arildno
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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.)

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

arildno
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The rule of thumb is to keep it simple.
An where, in these problem texts is the rule to apply Occam's Razor stated??

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.)

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.

berkeman
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:rofl: