Basic explanation and examples on mute variables I would like to get from you some feedback regarding mute variables. Specifically, I would love to get an easy definition or description, one that I could convey to high school kids. I would also need some examples, accordingly, to illustrate.
One last thing, a little bit more technical. I would love to know in which way mute variables in mathematics relate to mute variables in lambda calculus, as I know it from Logic. 
Bets regards and thanks in advance.
 A: You may think that a formula has an author and users. A formula can contain variables, that is, symbols that can be assigned values, very useful to parametrize. Such variables can be categorized by who, between the author and a user, is allowed to assign values.
$$f(x)=\sum_{i=1}^{10}x^i$$
Here the author has conveniently made use of the parametrization feature of variables with the variable $i$. She bound $i$ iteratively to the integers in the range $1$ to $10$. So $i$ is a bound variable.
At the same time the author gave the users of her formula the possibility to use that formula parametrically. That is, a user is free to assign a value to $x$, it's a free variable, it is open bound. Then for the user, the bound variable $i$ is not usable. It's dummy, so to say, from the user perspective, or it does not claim to be assigned, it's thus mute.
Moreover, as you say in your comment you can reuse the name $i$ in the same formula to represent different dummy variables, with no name clashing.
EDIT:
Here the same name $n$ is used to name two different dummy variables:
$$f(x) = \frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos (nx)+\sum_{n=1}^\infty b_n \sin (nx)$$
where
\begin{align}
a_0 &=\frac{1}{\pi}\int_{-\pi}^\pi f(x)dx\\
a_n &=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)dx\\
b_n &=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)dx\\
\end{align}
