So, what is the fourth number?

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Discussion Overview

The discussion revolves around identifying the fourth number in a numerical sequence defined by the initial values 6, 10, 14, and extending to 62 as the tenth number. Participants explore various rules and patterns that could generate such a sequence, examining the implications of these rules on the identification of specific terms within the sequence.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the fourth number is 22, based on a rule involving multiplication of 2 with prime numbers starting from 3.
  • Another participant agrees that the proposed rule works but emphasizes the existence of multiple sequences that could fit the given values.
  • A request is made for examples of other sequences that could produce the same values, leading to a suggestion of a periodic repetition of the first ten numbers.
  • Concerns are raised about whether the proposed periodic sequence adequately addresses the identification of the fourth number and other specific terms.
  • Discussion includes the notion that there can be infinitely many rules generating the same sequence, with examples provided to illustrate this point.
  • One participant argues that a sequence can be defined in various ways, leading to different interpretations of what constitutes the same sequence.
  • Another participant challenges the consistency of the proposed rules, indicating that different rules lead to different sequences.
  • Occam's Razor is mentioned as a principle suggesting that the simplest explanation may be the most useful, despite the existence of many possible solutions.

Areas of Agreement / Disagreement

Participants generally agree that there are infinitely many sequences that can match the given values, but they disagree on the implications of this for identifying specific terms within the sequence. The discussion remains unresolved regarding the validity of different rules and their outcomes.

Contextual Notes

Participants express uncertainty about the definitions and rules governing the sequences, highlighting that the lack of specificity in the sequence definition allows for multiple interpretations and patterns.

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the next sries:6,10,14...62
62 is the tenth number in the series.
what is the fourth number?

my answer is 22 because the series is a multplication of 2 with a prime number starting from 3.
2*3,2*5,2*7,2*11...2*31

so? is my answer correct?
 
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Your rule works ... so it must be correct!

Have confidence.
 
There exist an infinite number of sequences having the given values. Your rule works- that's all one can ask.
 
"There exist an infinite number of sequences having the given values."
can you show me the other sequences?
 
Originally posted by loop quantum gravity
can you show me the other sequences?

how about the sequence made from the first ten numbers repeated periodically?
 
Originally posted by Guybrush Threepwood
how about the sequence made from the first ten numbers repeated periodically?
care to explain?
 
2*3,2*5,2*7,2*11, ..., 2*31, 2*3,2*5,2*7,2*11, ..., 2*31, 2*3,2*5,2*7,2*11, ..., 2*31, 2*3,2*5,2*7,2*11, ..., 2*31, ...
and so on
 
but it doesn't solve for the fourth number in this sequence or the 14 24 etc.
 
Originally posted by loop quantum gravity
but it doesn't solve for the fourth number in this sequence or the 14 24 etc.

why not?
the series begins as you said 6,10,14...62
the tenth number is 62, the fourth is 22...
you didn't specified anything about the 14th or 24th number...
this is one of the infinite number of sequences HallsofIvy said (IMO)
 
  • #10
I think 'loop quantum gravity' just wants to know WHY there are infinitely many 'rules' that would produce a single sequence.

So, for a clear example, look at the sequence ...

2, 4, 6, 8, ...

Here are some 'rules' to describe it:

(a) {2n: n is a natural number}
(b) {2n/1: n is a natural number}
(c) {4n/2: n is a natural number}
etc.

Although there are infinitely many, all of them are just 'versions' of (a) and are NOT interesting or of any value.

Sometimes this is not the case, though, and two quite different 'rules' can produce the same sequence and both are 'interesting'.
 
  • #11
But there doesn't have to be a simple rule-

I can just declare the sequence to be 6,10,14, -300, pi, &sqrt;(2)
10000, 999, -1,62, 1, 2, 3, 4, 5,... with the rest being the positive integers in order.

I had a professor who gave us the sequence 21, 19, 17, 15, 13, and challenged us to find the next number.

The next number in the sequence was 32! Those were the numbers of the bus stops on his way to work. (Between 13 and 32, the bus turned off the main street it had been on.
 
  • #12
good point. I think Occam's Razor is the most relevant principle to this question. Granted, there are an infinite number of possible solutions, but chances are the most basic, logical answer is the most useful(though not of necessity the only right answer).


jeffceth
 
  • #13
Originally posted by Guybrush Threepwood
why not?
the series begins as you said 6,10,14...62
the tenth number is 62, the fourth is 22...
you didn't specified anything about the 14th or 24th number...
this is one of the infinite number of sequences HallsofIvy said (IMO)
but the rule is the same as i pointed out, multiplication of a prime number by two.
i thought that in a sequence there could be infinite patterns to it.
 
  • #14
Originally posted by loop quantum gravity
but the rule is the same as i pointed out, multiplication of a prime number by two.

no it's not because after your rule, the 11th number is 2*37 = 74 and after mine it's 6. they're 2 different series...

i thought that in a sequence there could be infinite patterns to it.

I don't quite understand this.
If you define a sequence as being 2*(succesive prime numbers) there's only one sequence.
But if you define a sequence as 6, 10, 14, ..., 62, ... there a a infinite number of sequences that match. If there's nothing more in the definition any sequence you find to match the 1st, 2nd, 3rd and 10th number as defined earlier is correct.
 

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