Adding subscripts This is a stupid question. But I'm trying to solve a Fibonacci problem and just realized that I don't know how to manipulate them. For example why does $F_{3n+1}$=$F_{3n-1}$+$F_{3n}$
 A: As I expect you know, any Fibonacci number is going to be the sum of the previous two. Now, if $F_x$ is the $x$-th Fibonacci number, then it's the sum of the previous two, namely $F_{x-2}$ and $F_{x-1}$.  
Now, in your problem, it isn't $x$ being used as a subscript, but $3n+1$. Okay. What would be the subscripts of two numbers immediately preceding? The one would be $(3n + 1)-1 = 3n$ and the one just prior would be $(3n + 1)-2 = 3n-1$  
Which is precisely what it was required to prove.
A: For Fibonacci numbers:
$$F_{u+1}=F_u+F_{u-1}$$
(The next term is the sum of the previous two)
Now let $u=3n$ to get what you want. The reason we can substitute values, is because $F$ is a function of $u$. For example if $F(u)=u^2$ than we can conclude $F(x)=x^2$.
A: In general, you cannot manipulate subscripts without knowing something about the function. $S_{n-1} + S_{n+1}$ does not equal $ S_{n+1+n-1} = S_{2n} $for an arbitrary series. However, the example you bring up can potentially be simplified, although that requires knowledge of what your end goal is. For a list of identities that can be used with the Fibonacci sequence you can reference...
https://en.wikipedia.org/wiki/Fibonacci_number
http://mathworld.wolfram.com/FibonacciNumber.html
Using identities found here you can manipulate series, and possibly even find closed form solutions for some problems.
