What's next in this number series?

Question

340, 680, 1428, 3141.6, _____

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

Answer 1

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

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

Answer 2

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 $p(4)=42$.

Answer 3

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!

Answer 4

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

Answer 5

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 $p(i)=a_i$.

Answer 6

7225.68

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

Answer 7

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.