1

Give a combinatorial proof to each of the Fibonacci identities: $$nF_0+(n-1)F_1+\dots\dots+2F_{n-2}+F_{n-1}=F_{n+3}-(n+2)$$ and $$ F_2+F_5+\dots\dots+F_{3n+1}=\frac{F_{3n+1}-1}{2} $$ Assume that $F_0=0, F_1=F_2=1, F_n=F_{n-1}+F_{n-2},\ \ \ for \ \ n\geqslant3$.

zaa
  • 708
user62452
  • 11
  • 2
    try http://math.stackexchange.com/questions/15469/combinatorial-proof-of-a-fibonacci-identity-n-f-1-n-1f-2-cdots-f-n?rq=1 – Yimin Feb 15 '13 at 19:06
  • 1
    Damn, combinatorial proof. – zaa Feb 15 '13 at 19:07
  • Please ask your second identity as a separate question. It also seems you mean $3 n - 1$ instead of $3 n + 1$ in the limit, perhaps the right hand side is also wrong? – vonbrand Feb 15 '13 at 20:32

0 Answers0