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Prove $f_{1} + f_{3} + f_{5} +...+f_{2n-1} = f_{2n}$ when $f_{n}$ is the $n$-th Fibonacci number.

Before I try to prove this, I am trying to understand this in English. What is happening in this equation? How can $2n-1 = 2n$? What is the English interpretation of this?

S.C.B.
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Dre_Dre
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    By $f_1+f_3+f_5+\dots+f_{2n-1}=f_{2n}$ They are saying $f_1+f_3=f_4$ and $f_1+f_3+f_5=f_6$ as well as $f_1+f_3+f_5+f_7=f_8$ etc... in general $\sum\limits_{k=1}^n f_{2k-1}=f_{2n}$. If it happens to be true, it will depend on what specifically the sequence $f_n$ is and the properties thereof. – JMoravitz Aug 12 '16 at 02:36
  • @JMoravitz For the inductive proof here is an example: it is true for n=1 F1 = F2 1 = 1

    suppose it is true for n F1 + F3 + F5 + ... + F2n-1 = F2n

    let's prove it for (n + 1) F1 + F3 + F5 + ... + F2n-1 + F2n+1 = F2n+2 (F1 + F3 + F5 + ... + F2n-1) + F2n+1 = F2n+2 F2n + F2n+1 = F2n+2

    How does he deduce that f1 = f2? How does F2n-1 + F2n+1 = F2n+2?

     – Dre_Dre Aug 12 '16 at 02:54
  • You tell me. What is $f_1$? What is $f_2$? (what is $f$ in the first place) Do those happen to be equal? – JMoravitz Aug 12 '16 at 02:57
  • @JMoravitz I am interpreting f1 and f2 as terms not numbers...terms in the sequence? – Dre_Dre Aug 12 '16 at 02:59
  • If they happen to be representative of the Fibonacci numbers, by definition $f_0=0, f_1=1, f_2=1$ and in general $f_n=f_{n-1}+f_{n-2}$ for all other values of $n\geq 2$. If it was a different sequence, then that might not necessarily be the case and the statement could be false. – JMoravitz Aug 12 '16 at 03:01
  • @JMoravitz ok let me understand this: fn is the nth term that we have not yet wrote down or who is next in the sequence, fn-1 is the previous number in the sequence and fn-2 is what? When I plug 1 into the formula n-1+n-2 I get 1-1 + 1-2 = -1 and I know this is not what it is saying. Pleas forgive me if it is obvious. – Dre_Dre Aug 12 '16 at 03:05
  • The Fibonacci sequence starts: 0,1,1,2,3,5,8,13,21,34,... (where $0$ is the "zero'th" number in the sequence) and it follows the property that every number in the sequence is the sum of the previous two numbers. (e.g. $34=13+21$. Also $21=8+13$ and $13=5+8$ etc..., so what number is next after $34$?). By $f_{n-1}+f_{n-2}=f_n$ we are saying exactly that property I mention, that if you add two consecutive numbers, you get the number that follows that. We are not saying that $n-1+n-2=n$ we are saying the $(n-1)$'st number plus the $(n-2)$'nd number is the $n$'th number. – JMoravitz Aug 12 '16 at 03:10
  • @JMoravitz Ok, got it! you are pure genius! – Dre_Dre Aug 12 '16 at 03:13

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