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

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

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 $p(4)=42$.

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!

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

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

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

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 $p(i)=a_i$.

7225.68

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