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340, 680, 1428, 3141.6, _____

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

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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 '13 at 16:01
@downvoter Could you please explain why this question is not appropriate rather than simply downvoting? –  Terry Li Feb 21 '13 at 16:01
Aptitude for what, may I ask? –  Marc van Leeuwen Feb 21 '13 at 16:08
@MarcvanLeeuwen It's for IBM jobs, to be honest. –  Terry Li Feb 21 '13 at 16:10
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 '13 at 17:47

7 Answers 7

up vote 152 down vote 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


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@Terry: You’re welcome. (That term close to $\pi$ is a nice red herring.) –  Brian M. Scott Feb 21 '13 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 '13 at 17:15
@Brian Out of curiosity, what was your thought process which led you to discover the pattern? –  jlund3 Feb 22 '13 at 2:45
@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 Feb 22 '13 at 9:48
The fourth Great Answer badge for Brian. STOP! –  Parth Kohli Feb 24 '13 at 6:24

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

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

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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 $p(i)=a_i$.

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To those unfamiliar with the notation the widehat above the terms means that those terms are omitted. –  Baby Dragon Feb 23 '13 at 20:18

340    * 2 = 680
680    * 2.1 = 1428
1428   * 2.2 = 3141.6
3141.6 * 2.3 = 7225.68
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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 $p(4)=42$.

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I would hope you'd get marks for writing this on the paper. –  Callum Rogers Feb 21 '13 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 '13 at 19:50
+1 for the 42 (obviously the right solution) –  Dominic Michaelis Feb 21 '13 at 20:45
This is a wonderful demonstration of why these kinds of questions are so silly. –  Plutor Feb 21 '13 at 22:20
I created a Mathematics account just to upvote this answer. –  asteri Feb 22 '13 at 19:50

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

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@Stefan Thanks for reformatting. It looks nicer now. –  user33040 Mar 22 '13 at 17:38

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!

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"Internet search gave me..." +1 –  Pedro Tamaroff Feb 21 '13 at 16:06
Always be honest ;) –  sonystarmap Feb 21 '13 at 16:07
+1 for Internet search. Didn't think about that. –  Terry Li Feb 21 '13 at 16:09
+1: Ostensibly, Google has improved all of our aptitudes :-). –  copper.hat Feb 21 '13 at 16:11
@copper.hat maybe it was Yahoo or Bing ;) –  sonystarmap Feb 21 '13 at 16:15

protected by Zev Chonoles Feb 22 '13 at 9:59

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