What's next in this number series?

[+58] [7]
Terry Li

[2013-02-21 15:57:59]

[
sequences-and-series
pattern-recognition
]

[ http://math.stackexchange.com/questions/310276/whats-next-in-this-number-series ]
[DELETED]

```
340, 680, 1428, 3141.6, _____
```

This is from an aptitude test. I'm not able to find any pattern in them.

[+160]
[2013-02-21 16:02:39]
Brian M. Scott
[ACCEPTED]

$\frac{680}{340}=2$, $\frac{1428}{680}=2.1$, and $\frac{3141.6}{1428}=2.2$, so we can expect that the person posing the question intended the next ratio to be $2.3$; this makes the next number

$$3141.6\cdot2.3=7,225.68\;.$$

(1)
oh, nice catch! - ** lhf**

clever. didn't thnk of it. - ** gt6989b**

This is probably the one. There's one option with the exact value. Thanks. - ** Terry Li**

(7)
@Terry: You’re welcome. (That term close to $\pi$ is a nice red herring.) - ** Brian M. Scott**

(44)
When giving us a question from a multiple-choice test, how about giving us the possible answers, as well? - ** GEdgar**

(2)
Dude, you're asking a math multiple choice question on math.sx? Maybe you shouldn't work for IBM =) - ** Rudie**

(6)
@Brian Out of curiosity, what was your thought process which led you to discover the pattern? - ** jlund3**

(2)
I came to the same answer independently; i'll say that my thought process was, 'Hmm, 340x2=680, I wonder if there is a relationship between 680 and 1428 - yep, 2.1...' and then just verified the rest. This is the sort of test they use to see if you have second-order thinking skills, so it is always a second degree relationship. - ** Joe**

(8)
@jlund3: By eye the growth was roughly but not quite geometric, so it was natural to look at the ratios of consecutive terms to try to get a better idea of just how the sequence was growing, and they turned out to be increasing in a very simple way. - ** Brian M. Scott**

(8)
The fourth Great Answer badge for Brian. STOP! - ** Parth Kohli**

1

[+116]
[2013-02-21 19:00:31]
user1354557

Although 7225.68 is the obvious answer, as mentioned in other solutions, it should be noted that there are an uncountably infinite number of "correct" answers which can be attained from 4th-degree polynomials. Just solve a linear systems of equations of

\begin{equation} p(x) = ax^4 + bx^3 + cx^2 + dx + e\end{equation} and \begin{equation} p(0) = 340,\;\; p(1) = 680,\;\; p(2)=1428,\;\; p(3)=3141.6,\;\,\text{and}\;\, p(4)=n \end{equation} where $n$ is any real number of your choice.

As an example, \begin{equation} p(x) = \frac{1}{60}(-15841x^4 + 100622x^3 - 178739x^2 + 114358x + 20400) \end{equation} produces an answer of $p(4)=42$.

(10)
I would hope you'd get marks for writing this on the paper. - ** Callum Rogers**

(4)
I'd go even further and say that practically any real is a correct answer, since for any five (or more) computable reals you can always find some sort of - more or less complicated - expression relating them (here, a 4th degree polynomial). But I suppose that would be perceived as missing the point and I'm not sure this would net you any points on the paper... sadly. - ** Thomas**

(52)
+1 for the 42 (obviously the right solution) - ** Dominic Michaelis**

(11)
This is a wonderful demonstration of why these kinds of questions are so silly. - ** Plutor**

(2)
bit.ly/ubitO4, for anyone who didnt know... - ** Matt Calhoun**

I tend to think that people of high aptitude would pass on this question if given more time on later ones. Awesome answer all the same! - ** meawoppl**

@MattCalhoun Wow, I never thought of it this way although I studied engineering and formulated impulse functions before. So now I know, I will write 42 for all the answers and then add that link at the end of the paper. I cannot be wrong. :) - ** Jake**

(8)
I created a Mathematics account just to upvote this answer. - ** asteri**

How did you even generate that function - ** Parth Kohli**

There may be uncountably many correct answers obtained from 4-th degree polynomials, but there are only countable many answers that are definable. - ** Donkey_2009**

2

[+55]
[2013-02-21 16:03:43]
sonystarmap

Internet search gave me 340, 680, 1428, 3141.6, 7225.68 as: \begin{align} 680/340 = 2 \\ 1428/680 = 2.1 \\ 3141.6/1428 = 2.2 \\ 7225.68/3.141.6 = 2.3 \end{align}

Edit: Or it could be a number on this
german webpage, which compares different types of ovens.
^{[1]} All other 4 numbers can be found there, so good look finding a pattern!

(16)
"Internet search gave me..." +1 - ** Pedro Tamaroff**

(13)
Always be honest ;) - ** sonystarmap**

(1)
+1 for Internet search. Didn't think about that. - ** Terry Li**

(9)
+1: Ostensibly, Google has improved all of our aptitudes :-). - ** copper.hat**

(3)
@copper.hat maybe it was Yahoo or Bing ;) - ** sonystarmap**

@macydanim: or DuckDuckGo. ;) - ** 2C-B**

Aw yis DuckDuckGo. (Yahoo might as well just go out of business already.) - ** Sean Allred**

3

[+8]
[2013-02-21 16:04:11]
user33040

$$ a_0 = 340, $$ $$ a_n = a_{n-1} \cdot \{2+0.1\cdot(n-1)\}. $$ So $$ a_4 = 3.141.6 \cdot 2.3 = 7225.68. $$

(1)
@Stefan Thanks for reformatting. It looks nicer now. - ** user33040**

4

[+3]
[2013-12-05 16:13:12]
Plop

The next term is 0. One can easily see that this sequence is defined by

$f : \left\{\begin{array}{cl} &0 \mapsto 340\\ &1 \mapsto 680\\ &2 \mapsto 1428\\ &3 \mapsto 3141,6\\ &n \mapsto 0 \mbox{ }\forall n \geq 4\end{array}\right.$

5

[+2]
[2013-02-23 16:54:49]
Baby Dragon

So I am going to echo some other answers and say that the next number can be any complex number. Let $$a_0,a_1,\cdots a_n$$ be any sequence of numbers. Their is a generic way of associating to this sequence, a polynomial $p(x)$ such that $p(i)=a_i$. One method is to simply solve a set the set of linear equations you get pluging $i$ into a generic $n+1$ degree polynomial. We can write the polynomial down directly however. To do this let us consider the following expression, $$\phi_{i,n}(x)=\frac{x(x-1)(x-2)\cdots \widehat{(x-i)}\cdots (x-n)}{(i)(i-1)(i-2)\cdots \widehat{(i-i)}\cdots (i-n)}$$. Note that $\phi_{i,n}(i)=1\mbox{ and }\phi_{i,n}(j)=1\mbox{ if }j=0,1,\cdots$ $(\widehat{i})\cdots n.$ Therefore, if we form the polynomial, $$p(x)=a_0\phi_{0,n}(x)+a_1\phi_{2,n}(x)+\cdots a_n\phi_{n,n}(x)$$. This polynimial has the property that $p(i)=a_i$.

(1)
To those unfamiliar with the notation the widehat above the terms means that those terms are omitted. - ** Baby Dragon**

6

[+1]
[2013-02-22 09:56:29]
Amit Modak

7225.68340 * 2 = 680 680 * 2.1 = 1428 1428 * 2.2 = 3141.6 3141.6 * 2.3 = 7225.68

7

gt6989bTerry Licopper.hatMarc van LeeuwenTerry Li`π/2`

and write up a function which generates exactly those first five elements. You can construct such a function easily with the use of the Dirac impulse. This proves the stupidity of such questions, which are not about any logical thinking, but about guessing what the examiner though about. Just like if they said "I though of a number, now guess which one is it.". -vszAlbert Renshawuser62947