# Sum of first n fibbonaci numbers

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• July 7th 2010, 05:50 PM
Chris11
Sum of first n fibbonaci numbers
Hey, I was just wondering if there was an explicit expression for the sum of the first n fibbonaci numbers.
• July 7th 2010, 06:05 PM
chiph588@
Quote:

Originally Posted by Chris11
Hey, I was just wondering if there was an explicit expression for the sum of the first n fibbonaci numbers.

$\sum_{i=1}^n F_i = F_{n+2}-1$

This can be shown by induction or by the explicit formula for the Fibonacci sequence.
• July 7th 2010, 06:54 PM
Chris11
Thanks.
• July 7th 2010, 07:01 PM
Also sprach Zarathustra
Proof:

$F_1=F_3-F_2$
$F_2=F_4-F_3$
$F_3=F_5-F_4$
.
.
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$F_{n-1}=F_{n+1}-F{n}$
$F_n=F_{n+2}-F_{n+1}$

now add them... what you get?