Find the 6th number

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madprocess
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Re: Find the 6th number

Postby madprocess » Wed Aug 13, 2008 7:28 pm UTC

I have
[math]f(a)_b = \sum_{k=1}^af(k)_{b-1} + \frac{ab(b+1)}{2}[/math]
with
[math]f(a)_0=f(0)_b=0[/math]
but I'm not sure how to get more general.

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Re: Find the 6th number

Postby jmorgan3 » Wed Aug 13, 2008 7:51 pm UTC

I have
Spoiler:
f(a,b)=f(a-1, b) + f(a, b-1) + 3(b-1).
This gives me right answers for all but f(52, 106). I get 1.2678 x 1043 for that one.

Does f(10,10) = 596228? If so, I'm probably just getting rounding errors.
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Doraki
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Re: Find the 6th number

Postby Doraki » Wed Aug 13, 2008 8:21 pm UTC

How can it be correct for f(1,1) ?

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Re: Find the 6th number

Postby jmorgan3 » Wed Aug 13, 2008 8:34 pm UTC

Looking at my code, it seems I hard-coded f(a, 1) = a, so I got f(1,1)=1 that way. I guess my series has one more special case than I thought.
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Re: Find the 6th number

Postby GreedyAlgorithm » Wed Aug 13, 2008 10:06 pm UTC

jmorgan3 wrote:Looking at my code, it seems I hard-coded f(a, 1) = a, so I got f(1,1)=1 that way. I guess my series has one more special case than I thought.

Too many special cases. f(10,10) =
Spoiler:
646580
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Re: Find the 6th number

Postby jmorgan3 » Wed Aug 13, 2008 10:47 pm UTC

But that's not in the original information. My sequence agrees with yours for the original numbers given once I give it the special cases (f(0,b)=0, f(a,0) = 0, and f(1,1) = 1).
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Re: Find the 6th number

Postby Doraki » Wed Aug 13, 2008 10:55 pm UTC

no, yours must have special case for f(a,1) and f(a,0) for every a since your formula says f(a,1) = f(a-1,1) + f(a,0) = f(a-1,1).

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Re: Find the 6th number

Postby jmorgan3 » Wed Aug 13, 2008 11:04 pm UTC

Right. I hadn't realized that. Still three special cases, though.
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Re: Find the 6th number

Postby GreedyAlgorithm » Wed Aug 13, 2008 11:44 pm UTC

jmorgan3 wrote:But that's not in the original information. My sequence agrees with yours for the original numbers given once I give it the special cases (f(0,b)=0, f(a,0) = 0, and f(1,1) = 1).

When I said "too many special cases", I didn't mean "this doesn't fit the information you know", I meant "this probably isn't the underlying cause of the information you know since underlying causes usually don't end up having arbitrary special cases". That's really the whole thrust of the argument that finding the next numbers in sequences is an excellent skill to learn.
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Re: Find the 6th number

Postby mike-l » Wed Aug 13, 2008 11:58 pm UTC

Edit:
Spoiler:
This seems to work:
nCr (a + b + 2, b) - nCr (b + 2, b)
Last edited by mike-l on Thu Aug 14, 2008 12:21 am UTC, edited 1 time in total.
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Re: Find the 6th number

Postby jmorgan3 » Thu Aug 14, 2008 12:00 am UTC

Sometimes they do have arbitrary cases, though. Your sequence arbitrarily defines cases for f(0,b) and f(a, 0). The Fibonacci sequence arbitrarily defines f(1)=f(2)=1. 0! =1. If I felt like it, I could probably make an expression with step functions that eliminates the special cases, but would that make it less arbitrary?
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Re: Find the 6th number

Postby GreedyAlgorithm » Thu Aug 14, 2008 3:32 am UTC

mike-l wrote:Edit:
Spoiler:
This seems to work:
nCr (a + b + 2, b) - nCr (b + 2, b)

And mike-l takes the gold. No special cases, nice and simple, just like what you'd expect nature to throw at you. Wait what? Wasn't it GreedyAlgorithm (me!) that came up with that example? Maybe it's just that he constructs sequences like that? The point is that if he's done his job well, then he's constructed a sequence like the sequences you'd expect to encounter on your own.

How did I decide on that sequence? Easy, I just took an example of a real live sequence I had to solve and twerked it slightly. You can find the original problem in this recently necro'd thread with the solution discussion here.

Now perhaps you'd like to argue that no one can really tell what sequences you're likely to encounter on your own, and so can't construct problems that don't require reading the writer. Hopefully not, though, because that seems like a very silly argument.

Aside: I actually started with sum(nCr(b+i+1,i+2),i=1..a), but as mike-l notes this simplifies to an even nicer form.
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Re: Find the 6th number

Postby Cosmologicon » Thu Aug 14, 2008 3:56 am UTC

Random832 wrote:
Cosmologicon wrote:a very similar thing comes up all the time in math. Haven't you ever seen something like this?

exp(x) = 1 + x + x2/2! + x3/3! + x4/4! + ...


Once it gets beyond the really obvious sequences like 1, 2, 3, 4 (the kind of sequence which you WOULD NEVER FIND ON SUCH A QUESTION) though, no....

I'd say that the integers, steps by a specific number (evens, odds, multiples of 3, etc), and maybe alternating signs, are just about all that's acceptable in a context like that (so, + ...2 - ...4 + ...6); anything else you need to be explicit about. I don't think you ever see a series written-out the way you did above that has "2 ....4 ....8 ....16" or "1 ....4 ....9 ....16" in it rather than 21 22 23 24 or 12 22 32 42

I think there are a few more cases that are okay to not be explicit, such as {0, 0, 1, 1, 2, 2, 3, 3,...}, {1, 2, 1, 2, 1, 2, ...}, and {2, 3, 5, 7, 11, 13,...}. In these cases it's also harder to write a formula for the nth term, so I think just starting the sequence is preferable. Furthermore, combinations of these slightly increase the complexity of the sequences you see. The sequence {1/3, 3/5, 5/7, 7/9, ...} is slightly more complicated than either {1, 3, 5, 7...} or {3, 5, 7, 9, ...}, even though it comprises them. So I think it's a good idea to expect people to recognize slightly more complicated sequences than the ones you name.

Anyway, even if the nth term is explicitly given, being able to get a feel for the sequence from the first few elements is still a valuable skill.

Also, you say that you'll "NEVER FIND SUCH A QUESTION" as an arithmetic sequence on a standardized test. Is that really so? It's my impression that standardized test questions along these lines are quite simple. Arithmetic and geometric sequences, mostly. Does anyone have any actual ones, or are we all just going off memory?

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Re: Find the 6th number

Postby mike-l » Thu Aug 14, 2008 5:26 am UTC

GreedyAlgorithm wrote:Aside: I actually started with sum(nCr(b+i+1,i+2),i=1..a), but as mike-l notes this simplifies to an even nicer form.


Spoiler:
I started with the recursion f(a,b) = f(a-1,b) + f(a,b-1) + b(b+1)/2, then worked it out as a triple sum, which I reordered and regrouped a few times to end up where we were.

Also, sums of n-gonal numbers, and sums of products of n and m-gonal numbers can usually be simplified by looking at pascal's triangle.
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Re: Find the 6th number

Postby mabufo » Thu Aug 14, 2008 5:53 pm UTC

Frimble wrote:
Token wrote:
Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

It irritates ME that people think "it could be anything" is a clever answer.


Really it comes down to whether they are testing understanding of mathematics or just how well the candidate understands the examiner.

The question should be: "Write down a formula that will generate this series." or better still they could relate it to properties of shapes, graphs or simple sets.


Forgive my lack of knowing, but how does one do that? Someone above your post mentioned some sort of formula, how does that work?

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Re: Find the 6th number

Postby Macbi » Sun Aug 17, 2008 1:42 pm UTC

I show you a question from a GCSE (age 15ish) paper I did reacently:

Code: Select all

Consider these sequences:
Q 1, 3, 5, 7, 9 ...
R 1, 4, 9, 16,25...
S 2, 8, 18,32,50...
T 1, 3, 9, 27,81...
U 1, 5, 9, 5,-31...

a) Write the next term in each sequence:
Q_
R_
S_
T_
U_

b) Write a general formula for each sequence:
Q_
R_
S_
T_
U_


Have a go at each of them before you look in the spoiler:
Spoiler:
Q 2n-1
R n2
S 2n2
T 3n-1
U S-T i.e. 2n2-3n-1


Okay. U would clearly not be guessable without the other questions, and was difficult to do with them. It was mearly a game of spot-the-gimmick. I'd put a quartic through those points before I found the right answer. I guess my point is that if you look for the simplest answer then your answer will be subjective, and if you try to work out what they were thinking of then you're doing science, not maths.
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Re: Find the 6th number

Postby Frimble » Sun Aug 17, 2008 2:52 pm UTC

mabufo wrote:
Frimble wrote:
Token wrote:
Frimble wrote:It irritates me that this sort of question frequently appears on 11+ tests and GCSE Maths exams. The next term could be any number at all, yet only one specific number is marked as correct.

It irritates ME that people think "it could be anything" is a clever answer.


Really it comes down to whether they are testing understanding of mathematics or just how well the candidate understands the examiner.

The question should be: "Write down a formula that will generate this series." or better still they could relate it to properties of shapes, graphs or simple sets.


Forgive my lack of knowing, but how does one do that? Someone above your post mentioned some sort of formula, how does that work?


I used the wrong word here. The correct term is function, and specifically they should ask for either an algebraic function of n which gives the nth term (an) of the series, (eg. f(n)=2n+3) or an iterative function which gives the nth term as a function of previous terms (eg. a1=5, an+1=an+2).

There are various methods of finding these functions. Some examples can often be determined by inspection, while others can require much more complicated methods.

Questions of this sort still have an infinite number of answers, but most of them would take a very long time to find.


When I mentioned relating the problem to 'shapes graphs or simple sets' I meant questions along these lines:

1) Given that the sum of the internal angles of a triangle is 180, the sum of the internal angles of a quadrilateral are 360 and the sum of the internal angles of a a pentagon are 540, express the sum of the internal angles of a polygon, a, in terms of the number of sides the polygon has, s.

2) James doing a 'dot to dot' puzzel in which he joins dots on a piece of paper with straight lines starting from the first dot and ending with the last dot. Given that one line is required to join two dots, two lines to join three dots and three lines to join four dots, express the number of lines, l required to join n dots in terms of the number of dots, n.

3) Sally is placing wooden beads on strings to make necklasses. She has two sorts of bead: square beads and round beads. She can make two types of necklass by placing one bead on a string, three types of necklass by placing two beads on the string, and six types of necklass by placing three beads on the string. How many types of necklass can she make by placing five beads on the string?


Obviously there is a problem with this sort of question too, in that it is testing reading comprehension as well as mathematical ability, but at least there is only one correct answer to these questions.

Does that answer your question?
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Re: Find the 6th number

Postby GreedyAlgorithm » Sun Aug 17, 2008 6:54 pm UTC

Macbi wrote:I guess my point is that if you look for the simplest answer then your answer will be subjective, and if you try to work out what they were thinking of then you're doing science, not maths.

Read Polya. Read Polya's How To Solve It. Then get back to me about "doing science, not maths".
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Re: Find the 6th number

Postby Frimble » Mon Aug 18, 2008 10:42 am UTC

GreedyAlgorithm wrote:
Macbi wrote:I guess my point is that if you look for the simplest answer then your answer will be subjective, and if you try to work out what they were thinking of then you're doing science, not maths.

Read Polya. Read Polya's How To Solve It. Then get back to me about "doing science, not maths".

What is your point here? That mathmatical problem solving is similar to the scientific method?
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Re: Find the 6th number

Postby Red Hal » Mon Aug 18, 2008 11:19 am UTC

Pathway wrote:3263442?

If so, 10650056950806, 113423713055421844361000442, 12864938683278671740537145998360961546653259485195806.


I had no idea that website existed. I feel so inadequate, yet insanely happy at the same time that I have a new bookmark! :D
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Re: Find the 6th number

Postby Random832 » Mon Aug 18, 2008 12:46 pm UTC

Cosmologicon wrote:
Random832 wrote:
Cosmologicon wrote:a very similar thing comes up all the time in math. Haven't you ever seen something like this?

exp(x) = 1 + x + x2/2! + x3/3! + x4/4! + ...


Once it gets beyond the really obvious sequences like 1, 2, 3, 4 (the kind of sequence which you WOULD NEVER FIND ON SUCH A QUESTION) though, no....

I'd say that the integers, steps by a specific number (evens, odds, multiples of 3, etc), and maybe alternating signs, are just about all that's acceptable in a context like that (so, + ...2 - ...4 + ...6); anything else you need to be explicit about. I don't think you ever see a series written-out the way you did above that has "2 ....4 ....8 ....16" or "1 ....4 ....9 ....16" in it rather than 21 22 23 24 or 12 22 32 42

I think there are a few more cases that are okay to not be explicit, such as {0, 0, 1, 1, 2, 2, 3, 3,...}


This is equivalent to 0, 1, 2, 3....

{1, 2, 1, 2, 1, 2, ...}


This almost certainly diverges.

, and {2, 3, 5, 7, 11, 13,...}. In these cases it's also harder to write a formula for the nth term, so I think just starting the sequence is preferable. Furthermore, combinations of these slightly increase the complexity of the sequences you see. The sequence {1/3, 3/5, 5/7, 7/9, ...} is slightly more complicated than either {1, 3, 5, 7...} or {3, 5, 7, 9, ...}, even though it comprises them. So I think it's a good idea to expect people to recognize slightly more complicated sequences than the ones you name.

Anyway, even if the nth term is explicitly given, being able to get a feel for the sequence from the first few elements is still a valuable skill.

Also, you say that you'll "NEVER FIND SUCH A QUESTION" as an arithmetic sequence on a standardized test. Is that really so? It's my impression that standardized test questions along these lines are quite simple. Arithmetic and geometric sequences, mostly. Does anyone have any actual ones, or are we all just going off memory?


Have you seriously had "Find the next number: 1, 2, 3, 4, _" on a test? Remember, the requirements I suggested for "allowed to be specified in a series without an explicit Nth term given" do not allow geometric sequences.


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