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$F_n= F_{n-1}+ F_{n - 2}$

I know that when a number is superscripted it means "to the power of", but what does the subscript mean?

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    http://en.wikipedia.org/wiki/Sequence2012-03-23
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    I understand that the equation denotes a sequence, I just don't know how to properly read the equation.2012-03-23
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    The subscript denotes what term of the sequence you are considering; first, second, third,..., $n$th,... In the instant case, $F_n$ is the $n$th Fibonacci number; $F_1$ is the first Fibonacci number, $1$; $F_2$ is the second Fibonacci number, $1$; $F_3$ is the third Fibonnaci number, $2$; etc. It is usually pronounced "sub $n$", or "sub $i$", or whatever the index is.2012-03-23
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    More precisely, we define $F_1=1$ , $F_2=1$ and from this $F_3=F_2+F_1$ and so on.2012-03-23
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    very insightful comments2018-11-22

4 Answers 4

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In this case the subscripts tell you which term of the sequence you’re looking at: $F_n$ is the $n$-th term of the sequence. This particular sequence is the Fibonacci sequence, which is defined by setting $F_0=0$ and $F_1=1$, thereby establishing the zero-th and first terms, and defining the rest recursively by the relationship that you quoted in your question: $$F_n=F_{n-1}+F_{n-2}\tag{1}$$ for all $n>1$. The formula $(1)$ then says that the $n$-th Fibonacci number is the sum of the $(n-1)$-st and $(n-2)$-nd Fibonacci numbers. When $n=2$, that says that $$F_2=F_1+F_0=1+0=1\;;$$ then when $n=3$ it says that $$F_3=F_2+F_1=1+1=2\;,$$ when $n=4$ it says that $$F_4=F_3+F_2=2+1=3\;,$$ and so on.

In this way we have an infinite sequence $\langle F_n:n\in\Bbb N\rangle=\langle0,1,1,2,3,5,8,\dots\rangle$. In general $\langle x_n:n\in\Bbb N\rangle$ is an infinite sequence $\langle x_0,x_1,x_2,x_3,\dots\rangle$, the subscripts indicating the position of each term in the sequence. In the sequence the order matters. That is, although the sets $\{x_0,x_1,x_2,x_3,\dots\}$ and $\{x_1,x_0,x_3,x_2,\dots\}$ are identical, the sequences $\langle x_0,x_1,x_2,x_3,\dots\rangle$ and $\langle x_1,x_0,x_3,x_2,\dots\rangle$ are not.

You can think of these subscripts simply as labels to keep the positions straight, just as we can use $\langle x_1,x_2,x_3\rangle$ for an ordered triple representing a point in $3$-space. From a more formal point of view, however, a sequence is actually just a function. For example, the sequence $$\langle x_0,x_1,x_2,x_3,\dots\rangle$$ of real numbers is a shorthand for the function $$x:\Bbb N\to\Bbb R:n\mapsto x_n\;,$$ so that we could just as well write $x(n)$ as $x_n$.

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    "In this case the subscripts tell you..." -- under what cases would the subscript mean something else?2014-06-03
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    @Brian M. Scott What does it mean when each subscript is non-numerical, and is exactly the same? I have an equation that has a value T (temperature in Kelvin), subscripted gamma. It's used several times each the same way, no variation, and all other equations in the same family simply use T, unsubscripted. What might that mean?2015-03-21
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    @DavidM: I don’t know; I suspect that one would have to know the notational conventions for the area of physics involved.2015-03-22
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Subscripts, like superscripts, have many uses. In this case, we have a function, probably with domain the non-negative integers or the positive integers, and $F_k$ is an abbreviation for $F(k)$.

So you can read the equation as $F(n)=F(n-1)+F(n-2)$. The value of the function at $n$ is the sum of the function values at $n-1$ and $n-2$.

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$F_n$ is the $n$-th term of the sequence (Fibonacci sequence, I suspect that's your case). For example $F_1=1$, $F_2=2$, $F_3=3$, $F_4=5$, $F_5=8$ and so on. The equation $$F_n=F_{n-1}+F_{n-2}$$ means that the $n$-th term of the sequence is obtained by the sum of the 2 immediately previous terms. Usually the subscript are a natural number, i.e. a number in $\{1,2,3,\ldots\}$. That means that the equation above only make sense if $n\geq 3$. This means that in order to describe a sequence by the above equation you must specify the values $F_1$ and $F_2$. For example you can verify that $$F_1=1,\space F_2=1,\space F_n=F_{n-1}+F_{n-2}$$ describes the Fibonacci's sequence.

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Something like $F_8$ usually means the $8$th term in a sequence called $F$. (And similarly for other numbers than $8$ . . . . . . .)