Proof Without Words: Fibonacci Triangles and Trapezoids

Proof Without Words: Fibonacci Triangles and Trapezoids

55
VOL. 86, NO. 1, FEBRUARY 2013
Proof Without Words:
Fibonacci Triangles and Trapezoids
Á N G E L P L A Z A
University of Las Palmas de Gran Canaria, Spain
[email protected]
HANS R. WALSER
Mathematisches Institut Uni Basel,Switzerland
[email protected]
I. Counting triangles:
fn
fn
2
fn
II. Identity:
n
X
fn
2 fn
2
fn
2
2 f k2 + f n2 + f n+1
= ( f n + f n+1 )2 .
k=1
1
f
1
f
2
f3
fn2
fn
fn 2+1
fn
2
fn +1
( fn + fn +1 )
P
The identity proved here reduces algebraically to nk=1 f k2 = f n f n+1 . This proof
uses almost the same diagram as [1] to prove a different identity.
REFERENCE
1. Hans R. Walser, Proof Without Words: Fibonacci Trapezoids, Math. Mag. 84 (2011) 295. http://dx.doi.
org/10.4169/math.mag.84.4.295
c Mathematical Association of America
Math. Mag. 86 (2013) 55. doi:10.4169/math.mag.86.1.055. This content downloaded from 193.145.138.9 on Mon, 9 Sep 2013 06:03:39 AM
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