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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 $F_{3n+1}$=$F_{3n-1}$+$F_{3n}$

ematth7
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  • What does it mean to "take apart things with subscripts"? It's not clear what you're asking. – Brick Sep 14 '15 at 22:02
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    I think he means "manipulate expressions involving subscripts". There might, for example, exist a sequence where the relationship between terms is $S_n+S_{2n}=S_{3n}$ or where an expression like $T_{n-1}+T_n=3T_{n-2}. – tomi Sep 14 '15 at 22:08
  • We're going to need the specific expression you are interested in simplifying and any thoughts you have on what you could do. It is possible to write $F_n$ as an expression in terms of $n$... – tomi Sep 14 '15 at 22:11
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    So sorry for the confusion!

    I mean't like why $F_{3n+1}$=$F_{3n-1}$+$F_{3n}$

     – ematth7 Sep 14 '15 at 22:55
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    The Fibonacci numbers are defined to satisfy $F_{k - 1} + F_k = F_{k + 1}$ (plus initial conditions on what $F_1$ and $F_2$ are). Just use $k = 3n$. – pjs36 Sep 14 '15 at 23:02

3 Answers3

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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.

MathAdam
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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 $F(x)=x^2$.

Ahmed S. Attaalla
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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.

Brevan Ellefsen
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