Meaning of the term $x_n$ I've started the lesson an Newton's method, and am Looking at this formula, 
$$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$$
I know when the numbers are to the top like in $x^2$, what that is, but what does it mean when they are subscripted, to the bottom like in the Newton's method formula?
 A: A subscript, as $x_n$, often means the n-th term in a sequence.The sequence may be finite or infinite and it is not necessary to start at $n=1$ nor at $n=0$.For example "let p_n be the n-th prime " (in increasing order,which in this case is usually not stated but assumed) means that (assuming we start the count at $n=1$) $p_1=2,p_2=3,p_3=5,p_4=7$,etc. In calculus we are almost always talking about  sequences, and it often more readable to use subscripts.For example if we have a sequence of FUNCTIONS we could write them as $f(1),f(2),...$,etc but then when $x$  belongs to the domain of the function $f(1)$ then the value of this function at the point $x$ is $f(1)(x)$ because the 4 consecutive keystrokes "$f(1)$" comprise the name of the function.It is easier to read if we write $f_1,f_2,...$ and then write $f_1(x)$ for the value of $f_1$ at $x$. Subscripts are not limited to natural numbers, and a subscript may have its own subscript...Example :Theorem: For each real number $x$ let $V_x$ be an open interval containing $x$. Then there exists an infinite sequence $x_1,x_2,x_3,...$ of real numbers such that for every real $x$, there exists at least one $n$ for which $x\in V_{x_n}$.
A: Newton's method seeks to find an approximate solution to the equation "$f(x)=0$" (i.e. a root of $f$).
$x_0$ is an initial "guess" at a solution. Feeding this into your equation: $x_1 = x_0 - f(x_0)/f'(x_0)$ gives a better approximate solution. 
Feeding $x_1$ into the equation again: $x_2=x_1-f(x_1)/f'(x_1)$ gives an even better approximate solution.
$x_n$ is the $n$-th attempt (i.e. iteration) at finding a solution. 
