67.
9 Fibonacci Matrices
Author(s): Sam Moore
Source: The Mathematical Gazette, Vol. 67, No. 439 (Mar., 1983), pp. 56-57
Published by: Mathematical Association
Stable URL: http://www.jstor.org/stable/3617368
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56 THE MATHEMATICAL GAZLEIE
which is valid for all x > -1 (and indeed for all unreal complex x). What is
more, the finite continued fractions obtained by terminatingthis expression
at any stage are the Pade approximantsto log (1 + x). (See, for example.
P. G. Dixon's book reviewin the December 1981 Gazette.)
The first few of these finite continued fractions agree, by simplification,
with those obtained in the inequalities above, although at first sight they
look different. The inequalities could, in fact, be deduced from the infinite
continuedfraction, supposingthat it had first been established!
I am not, of course, recommendingthat continued fractions be revived in
schools. But I have briefly outlined a fascinating experimentwhich can be
offered in the classroom with some prospect of student participation, and
which happens to tie in with the venerable subject of continued fractions. A
similarproject with tan-' x is also feasible.
E. R. LOVE
Department of Mathematics, University of Melbourne, Victoria 3052,
Australia
67.9 Fibonacci matrices
/I 1\
If M is the matrix ( ,then
M2 2 3\ /5 8 13 21
M2
3 '5
), M3-
8
M4) 21
13, 341 2
and the occurrence of the Fibonacci sequence
1, 1,2,3,5,8, 13,21,...
(and the reason why it occurs) is clear.
If we multiply each of these matrices by a factor in order to make the
leading entries 1 we get
(; D I )( )**
Because of the well-knownpropertiesof the ratios of successive terms in the
Fibonacci sequence, it is clear that this sequence of matrices is converging
to
( 1+) or ( 2)
where r is the Golden Ratio, ?(1 + /5).
It is an interesting conjecture that if one starts with the matrix
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NOTES 57
1 \
I(1t ), forms powers of it, and scales them to make the leading entries
1, then the sequence approaches
I 0
where f = i(x + /(X2 + 4)). I've confirmed with calculating aids that, for
severalx, this certainly seems the case.
SAM MOORE
CommunityCollege of Allegheny County,808 Ridge Avenue, PittsburghPA
15212, U.S.A.
67.10 Those infinitesimultaneousequations
In note 66.3 (March 1982) it was shown that the set of equations
X1 - X2 - X3 - X4 -... =0
X2 - X3 - X4 - =0 (*)
X3 -,
..X =0
had the non-trivialsolution xr = 1/2r- and the question was asked "where
did that come from?"
Here are two possible answers.
(1) If one tries xr = XoAr,it turns out that A= ?.
(2) If one takes a generatingfunction
0 tr- 1
(t) xr ( **)
r=1 (r 1)
of the x, x2, x3, ... it can be seen that x(t) = et2.
This is the unique solution of the integralequation
00
2x(t) + lim exp (-au2 + u)x(t - u)du = 0
a- Jo
which any function x satisfying (*) and (**) must satisfy. The mathematics
is perhapstoo technical to include here, but the comment is made in orderto
bringthis techniqueto the attentionof interestedreaders.
LL. G. CHAMBERS
School of Mathematics and ComputerScience, University College of North
Wales, Bangor LL57 2 UW
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