# Find the next term in the sequence and explain why

For one of my homework problems, I got a sequence that says 80 74 63 53... It also says that I need to find the next number in the series and what the sequence is. Either this is really hard, or the teacher made a mistake. Answer the question and you have my thanks.

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What did you try? What is your outcome vs. that by the teacher? –  abiessu Oct 2 '13 at 21:33
I tries to go online and searched up 'sequence finder'. Then I got a bunch of results. I typed in the sequence, but it either said it was impossible or irrelevant. –  danilka1 Oct 2 '13 at 21:35
Also, what is the question? We can't answer it if you don't give it. –  anorton Oct 2 '13 at 21:35
You can continue the sequence however you like. As long as the next term is $42$, of course. –  Daniel Fischer Oct 2 '13 at 21:35
@DanielPalamarchuk There are infinitely many rules that produce any finite start of a sequence. You can invent any rule to continue however you like. If you are given some constraints on the type of rule, that might make it unique. If the rule must be given by a polynomial of degree at most $3$, for example, that would determine it. –  Daniel Fischer Oct 2 '13 at 21:46

Here is how to apply finite difference methods to obtain further numbers in a sequence that is based on a polynomial:

80
= -6
74     = -5
= -11    =  6
63     = 1
= -10
53


In the above "difference tree", the first two numbers $80$ and $74$ are subtracted; since $74$ is the second term, $80$ is subtracted from it to get $-6$. Continue the process downwards taking the $3$rd term minus the $2$nd term and so on, to obtain the $-11$ and $-10$. Then apply another round of differences, so $-11 - (-6)=-5$ for the first element of the third column, and the other element is $1$. The final column is the last available difference, which is $1-(-5)=6$.

Once the initial tree is created, then every element of the sequence can be created using sums:

80
= -6
74     = -5
= -11    =  6
63     = 1
= -10   =  6
53     = 7
= -3
50


First, duplicate the final column downwards as long as necessary. Then, for each missing item (vertically), add the number immediately above to the number up and to the right, so we get $1+6=7$ as the third element of the third column, and $-10+7=-3$ and finally $53+-3=50$.

This process creates a polynomial $p(x)=80-6x-5{x\choose 2}+6{x\choose 3}$. This process only applies if you can assume that all the values in the sequence match a given polynomial. It has no bearing on the following sequence (for example):

1
11
21
1211
111221


Note that the above "difference tree" would allow you to choose any value to be the next value in the series, at a cost of having a degree $4$ polynomial. Any string of numbers can be formed into a polynomial sequence in this manner.

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Thank you for the answer –  danilka1 Oct 2 '13 at 21:48
Sorry to throw the spanner, but $74 - 63 = 11$. –  Daniel Fischer Oct 2 '13 at 21:55
@DanielFischer, it's supposed to be 63-74, not 74-63. –  danilka1 Oct 2 '13 at 21:57
@DanielPalamarchuk Yes, but that still makes it $-11$ and not $-9$. (Small mistake, nothing serious, the principle is good) –  Daniel Fischer Oct 2 '13 at 21:59
hmmm... I suppose you're right. –  danilka1 Oct 2 '13 at 22:00

hi i am a 7th grader and also received this problem last night. I have 2 Daniels in my class, so not sure if we are working on the same problem! I simply tried some regression analysis and this is what I came up with:

let's assume the outputs are 80,74,63,53 and so on.

let's assume we have 2 variable inputs. say x and z. the x inputs that correspond to the outputs above are 9,8,7,6 and these continue to descend.

let's assume the z inputs are -1, +2, 0 and these are repeating.

if you then use the function y=9x+z I think the next term in the pattern would be 47.

I suspect there is not a unique solution to this problem, so this should be an acceptable solution given the lack of constraints on the function.

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