What's next in this number series?

340, 680, 1428, 3141.6, _____


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

-
Not sure. The last one is roughly $1000\pi$, the one before is roughly $1000e$, and the second is double the first, that's all I can see... – gt6989b Feb 21 at 16:01
@downvoter Could you please explain why this question is not appropriate rather than simply downvoting? – Terry Li Feb 21 at 16:01
Maybe is it is not quite 'guess the number'? – copper.hat Feb 21 at 16:02
Aptitude for what, may I ask? – Marc van Leeuwen Feb 21 at 16:08
For such questions I always have the urge to answer 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.". – vsz Feb 21 at 17:47

$\frac{680}{340}=2$, $\frac{1428}{680}=2.1$, 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

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

-
@Terry: You’re welcome. (That term close to $\pi$ is a nice red herring.) – Brian M. Scott Feb 21 at 16:05
When giving us a question from a multiple-choice test, how about giving us the possible answers, as well? – GEdgar Feb 21 at 17:15
Dude, you're asking a math multiple choice question on math.sx? Maybe you shouldn't work for IBM =) – Rudie Feb 21 at 22:47
@Brian Out of curiosity, what was your thought process which led you to discover the pattern? – jlund3 Feb 22 at 2:45
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 Feb 22 at 5:36

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

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

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

-
I would hope you'd get marks for writing this on the paper. – Callum Rogers Feb 21 at 19:15
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 Feb 21 at 19:50
+1 for the 42 (obviously the right solution) – Dominic Michaelis Feb 21 at 20:45
This is a wonderful demonstration of why these kinds of questions are so silly. – Plutor Feb 21 at 22:20
I created a Mathematics account just to upvote this answer. – Jeff Feb 22 at 19:50

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. All other 4 numbers can be found there, so good look finding a pattern!

-
"Internet search gave me..." +1 – Peter Tamaroff Feb 21 at 16:06
Always be honest ;) – macydanim Feb 21 at 16:07
+1 for Internet search. Didn't think about that. – Terry Li Feb 21 at 16:09
+1: Ostensibly, Google has improved all of our aptitudes :-). – copper.hat Feb 21 at 16:11
@copper.hat maybe it was Yahoo or Bing ;) – macydanim Feb 21 at 16:15

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

-
@Stefan Thanks for reformatting. It looks nicer now. – Andrew Tuggle Mar 22 at 17:38

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 $p(i)=a_i$.

-
To those unfamiliar with the notation the widehat above the terms means that those terms are omitted. – Baby Dragon Feb 23 at 20:18
7225.68

340    * 2 = 680
680    * 2.1 = 1428
1428   * 2.2 = 3141.6
3141.6 * 2.3 = 7225.68
-

Most of such aptitude test questions can be answered based on forming a pattern by looking at the given data. Here it is very obvious that 680 is twice 340 and is in the second position so the number 2 has importance. Then when you look at the number in the third place you can fairly assume that it cannot be 3 times 340 or 3 times 680. That means there is a decimal at play or atleast you can rule out that the next number is a product of multiplying the index location with the number before.So what Brian M. Scott says above is true. But all you really had to do was get hold of a calc and take some random shots at it. Most aptitude tests are truly childs' play.

-
It's not an integer, so it must be a decimal fraction. Elementary mathematics! -1 – Potatoswatter Feb 22 at 1:10
I wonder why the down vote ? >>That means there is a decimal at play – happybuddha Feb 25 at 22:13

protected by Zev ChonolesFeb 22 at 9:59

This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation on this site.