relevance of number sequences?

Question

1, 4, 9, 16, 25, ?
of course, everyone (well, almost) would say "these are just the square-numbers; the next one is 36".

but as a mathematician you know that every number sequence
a_1, a_2, a_3, a_4, a_5, ...
has a infinite number of functions f which satisfy
f(1)=a_1, f(2)=a_2, f(3)=a_3, f(4)=a_4, f(5)=a_5. the values f(6) can be different as well. you could even conclude, that every number x is a continuation of the sequence, because you could find a function satisfying f(6)=x.

therefore there is not a "single" correct answer to number sequences.

now you could say: "you should always find the easiest/most simple function of all these". but what do you define as simple?

as a mathematician you would say "its very easy to calculate _polynomials_ to solve number sequences" (you just have to solve a system of linear equations).

so the main question is: how should you evalue/rate possible solutions to number sequences? or should you rather rate the corresponding function found?

Answer 1

As a math teacher, your question has significant meaning when discussing sequences and series--and the entire notion of "pattern hunting." I typically instruct my student to avoid exercises involving pattern hunting unless the pattern is "obvious." So, if we've just spent an hour talking about geometric series and I ask them to add up

1+(1/3)+(1/3)^2+(1/3)^3...

I would expect them to recognize that this is geometric with a=1 and r=1/3.

On the other hand, I probably wouldn't expect my average student to notice that the next term in the sequence [the Ulam sequence for (1,5)]

1,5,6,7,8,9,10,12,20,...

is 22.

Thus, the evaluation should certainly be in the context of the level of the mathematical discussion taking place and in the classroom, in regards to how the material will be used later.

Answer 2

Great question! I have spent a great deal of time studying number sequences and I completely understand where you are coming from.

Typically, the best answer is a formula that can be determined by looking at the first few numbers and then when applied, describes all the given numbers in the sequence.

It is important to learn about sequences because it helps one learn how to identify certain types of growth in functions and also apply to the real world as often people have to use mathematical models to describe real world events.

If I could wave a magic wand to make it so, all number sequence puzzles would have at least 6 given terms and should be determinable without the last two or three numbers given.

Answer 3

It is the corelation you find to be suits for this type of sequences. Here it is nothing but the squares of the numbers 1, 2, 3, 4 & 5. It is also very right that the next number is six square, i.e. 36.

Answer 4

I have (being a student) discussed this, or at least attempted to discuss this, with my teacher, and he amazingly does not even appear interested in discussing it...

Anyway, my opinion is:

The formula should be rated using "common sense" (whatever that be), and the one which appears to be most related to the problem should be used.

Of course, when posing a question involving the determination of a formula, background information should be given. Then results and graphs should match the background information.
If no background information is given, the question is purposeless.

Answer 5

I always want my series to either match up with something in the real world (e.g. the number of unique neckaces of black and white beads of length N) or something in mathematics (the number of primes of N digits).

Giving some monster polynomial when another answer is "the number of letters in the months of the year" is maybe 1 point for effort.

Answer 6

I would think it depends on the purpose of the calculation. Is it desired to know how the numbers are related or to generate the string. It might even just be so that you can extract data from the string. these all generate different requirements. What might be required out of your sequence could be 3,5,5,9 and then what?

Answer 7

Man, mathematics is just a tool to help us solve problems in other practical studies. If you get too involved into mathematics you'll go nuts!

Answer 8

Don't be a smart Alec when answering questions like that. All that'd happen to you is you getting no marks.

Answer 9

Perfect expectations of number sequeces, that you can't do with people.