3 7 14 23 36 49 66 83 What number comes next?

  • Context: High School 
  • Thread starter Thread starter BicycleTree
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Discussion Overview

The discussion revolves around identifying the next number in the sequence: 3, 7, 14, 23, 36, 49, 66, 83. Participants explore various mathematical approaches and theories related to the sequence, including polynomial roots and summation of squares.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that the numbers are the smallest 8 roots of a 9th degree polynomial whose 9th root is 123456789.
  • Another participant hints that extending the sequence could make it easier to identify a pattern, particularly by examining the 10th and 20th terms.
  • A participant questions the correctness of the 7th term being 67, stating their formula generates 66 for that term and 104 for the 9th term.
  • Another participant agrees with the assumption that the next term is 104.
  • One participant acknowledges a mistake and confirms that 104 is indeed the answer.
  • A request is made for an explanation of how to derive the next term.
  • An explanation is provided that the sequence is related to the sum of the squares of integers with corresponding prime numbers.

Areas of Agreement / Disagreement

There is some disagreement regarding the correctness of the 7th term and the method to derive the next term. While some participants assert that 104 is the next term, others question the validity of the earlier terms in the sequence.

Contextual Notes

Participants reference different methods and formulas, indicating potential limitations in their approaches, such as dependence on specific definitions or assumptions about the sequence.

Who May Find This Useful

Individuals interested in mathematical sequences, polynomial roots, and number theory may find this discussion relevant.

BicycleTree
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Probably not easy:

3 7 14 23 36 49 66 83

What number comes next?
 
Last edited:
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These are the smallest 8 roots of a 9th degree polynomial whose 9th root is 123456789
 
It's not quite that arbitrary.

Here's a hint: if I continued the sequence for 20 more terms then it would be much easier to figure out. In fact, from looking at the 10th and 20th terms you could get it almost immediately.
 
Last edited:
Are you sure 67 is correct for slot 7? I have a formula that gives all of them except it generates 66 for the 7th term.

EDIT: To check, my formula generates 104 for the 9th term.
 
Last edited:
I agree with Moo, and assume the next term to be 104 [in white].
 
You're right, I did make a mistake. 104 is the answer! Congratulations.
 
Last edited:
is someone going to show how to do it?
 
Explanation in white: It's the sum of the squares 1, 4, 9, ... with the corresponding primes 2, 3, 5, ...[/color]
 

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