Find the incremental rule of a series of values

[CheertheSecond]

This is the only forum I am attending at the moment so I don't have any other place to ask this.

So I got a series of values
125
350
800
2000
8000

Is there a formula to find the incremental rule?

 

[TsumiHokiro]

The On-Line Encyclopedia of Integer Sequences (OEIS)

 

[prognastat]

Well at least it isn't just me that couldn't find one. No matter what I try it breaks after the 800.

 

[CheertheSecond]

Well at least it isn't just me that couldn't find one. No matter what I try it breaks after the 800.

Wait! So there are sequences that have no rule?

 

[TsumiHokiro]

Of course there are. Their rules are not mathematically defined in that case.
They are just a sequence of numbers. You cannot figure out which would be the next member of the sequence.

 

[CheertheSecond]

Of course there are. They are not mathematically defined in that case.

I thought with all the numbers, we could have a result/ an answer.

So mathematically even if there is hypothetically infinite universes with each slightly different from each other, there are still things that could not exist?

 

[TsumiHokiro]

So mathematically even if there is hypothetically infinite universes with each slightly different from each other, there are still things that could not exist?

As the wiki tells us: In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. So not every sequence of number can make what is called a "sequence" in mathematics.

If it is a sequence, even though you are usually given only the first few numbers in it, 5 or 6, it actually has infinite numbers. And it has a rationale behind it which allows you to discover whichever number in the sequence you desire. It's not simply any number in any order. There is, however, in the infinite universe of numbers, an infinite number of sequences. It's a matter of whether they are significant to us or not.

 

[LilRora]

This is mostly guesswork, unless you are told it's supposed to be a geometric or algebraic or some other specific series. It's definitely not algebraic or geometric though, and I don't see any other good simple solution.

It's probably some more complicated series, where it's impossible to tell how the value changes without detailed analysis.

Also, I want to say this, I'm pretty sure that technically any finite sequence of numbers can be represented with an equation or a series. In cases of infinite sequences it's a bit more complicated, but the thing is, you are never given an infinite sequence; you are given a finite one that, if you find a formula that matches it, you can extend to infinity.

It's just that in many cases finding that equation or series is practically impossible and completely impractical, plus the fact that such a thing can be found doesn't mean the sequence was written with the intention to match one.

 

[RiaCorvidiva]

I agree that there probably isn't a single, objectively correct pattern, but if we do a bit of factoring, we'd get:

125 = 5 * 5 * 5
350 = 5 * 7 * 10
800 = 8 * 10 * 10
2000 = 10 * 10 * 20
8000 = 20 * 20 * 20

If there were some sort of pattern, we might be able to suggest

22400 = 20 * 28 * 40
51200 = 32 * 40 * 40
128000 = 40 * 40 * 80
512000 = 80 * 80 * 80

And while that is a real poor excuse for a pattern, I think that's definitely one of the more defensible continuations you could make.

 

[CheertheSecond]

I agree that there probably isn't a single, objectively correct pattern, but if we do a bit of factoring, we'd get:

125 = 5 * 5 * 5
350 = 5 * 7 * 10
800 = 8 * 10 * 10
2000 = 10 * 10 * 20
8000 = 20 * 20 * 20

If there were some sort of pattern, we might be able to suggest

22400 = 20 * 28 * 40
51200 = 32 * 40 * 40
160000 = 20 * 20 * 40
640000 = 40 * 40 * 40

And while that is a real poor excuse for a pattern, I think that's definitely one of the more defensible continuations you could make.

I don't get it. What would be an acceptable rule for this sequence?

 

[RiaCorvidiva]

Mostly I just made one up that sort of fit the pattern, and, doing it at 4 AM just after waking up, I made an error that I had to correct.

In this case, rather than a definitive formula for the nth term, we have to sort of define it recursively

a1 = 125
a2 = 350
a3 = 800
a4 = 2000

a(n+4) = a(n_) * 64

Again, it's a real ad hoc way of going about it, but that's what just sort of intuitively makes sense to me as a first pass.

Also, apparently (n_) without the underscore autoconverts into a thumbs-down emoji.

 

[Seren]

based on the series of values you have provided, the task is to find the incremental rule or pattern that governs the sequence. by specifying this formula, you can generate subsequent terms in order without listing them one by one. you start by; examining the given series 125, 350, 800, 2000, 8000. ...it appears that each successive term is formed by multiplying the previous term by a certain factor. by deducing this factor, you will be able to establish the incremental rule that governs the sequence. to determine the factor, you can examine the relationship between each number in the series. the first term of 125 is multiplied by a factor of 2.8 to yield a second term of 350. continuing pattern, the second term is multiplied by approximately 2.286 to produce a third term of 800. to the similar, the third term, 800, is multiplied by 2.5 to obtain the fourth term of 2000 and then finally, the fourth term of 2000 is multiplied by 4 to yield the fifth and final term of 8000. from this analysis, we can conclude that the incremental rule of the series is to multiply each term by a certain factor. this factor increases with each progression in the sequence, starting with 2.8, then 2.286, followed by 2.5, and finally 4.

however, it is important to note that the given data points are limited, and we cannot determine the incremental rule with absolute certainty based solely on these terms. to confirm the rule and its accuracy, just gather additional terms in the series and observe if they align with the rule determined through the preliminary analysis. to summarizing, the incremental rule for the given series appears to involve multiplying each term by a factor that increases with each progression. specific values for the factor are derived based on the correlation between successive terms in the sequence

 

[Zirrboy]

I don't get it. What would be an acceptable rule for this sequence?

That's the issue mentioned across most of the posts. There isn't really any restriction, so the measure of reasonability is subjective.

Entries obtained by simple calculation (ie + - * / powers, factorials...) and recursion (reference to previous elements) would be good candidates in most cases, but since recursive reference requires predefined starting elements you can always treat more elements as predefined until your scheme fits.

For example you can use the following expression to generate the latter two, if you prepend 100 it also fits for the second, but whether you find that acceptably brief/simple is up to you.

a_(n+1) = 2 * a_n + a_(n-1) + 25 * (n-2) * (-1)^n

a_0 = 100, a_1 = 125, <- these are predefined to make the rest work, so this approach treats them as arbitrary
a_2 = 2 * 125 + 100 + 0 = 350,
a_3 = 2 * 350 + 125 - 25 = 800,
a_4 = 2 * 800 + 350 + 50 = 2000

With that, the following numbers would be 4725(which would have been 8000 in your sequence), 11550, 27700, 67100, 161725,...

But things like "the number of holes in the characters of the Arabic decimal representation" can be taken just as well, and since you only have the beginning of the sequence, any apparent scheme could break at some point. https://oeis.org/A061419 for example starts much like the Fibonacci sequence, but quickly diverges, following a subjectively uglier formula.

 

[justabot]

This is the wrong platform for this type of thread. Locked.