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of symbols, can define general semantic concepts which are the basis of any no-
tion of logical validity (truth in all possible worlds, according to a definition due to
Leibnitz).
5.6
Series and Growths
Fibonacci sequence is related to a growth process. Growth processes occur in any
life phenomenon and at each level from the cell to the most complex organisms, and
to biological populations. The mathematical form of a growing process is very often
given by a series, that is, a sequence where at each step an additive term is added to
the term which is the outcome of the preceding step.
5.6.1
Arithmetical Progressions
The simplest form of growth is represented by a constant additive increment. It is
typical of a population where the number of individuals increases at each step by
a constant number of
k
individuals. If the initial quantity is
F
(
0
)
then the formula
which provides the population size at step
n
+
1 is given by:
F
(
n
+
1
)=
F
(
n
)+
k
that, by a trivial induction argument, gives for
n
>
0:
F
(
n
)=
F
(
0
)+
nk
.
If the population, at each step
i
increments of
ki
units, then, the formula providing
the size
G
(
)
(
)=
n
of such a population, by assuming that
G
0
0, is:
n
i
=
1
F
(
i
)=
n
i
=
1
ik
G
(
n
)=
The value
G
(
n
)
is thus:
ยท
(
+
)
k
n
n
1
G
(
n
)=
.
(5.2)
2
This equation can be easily proved by induction. Without loss of generality we con-
sider the case
k
0,
therefore the formula holds for the value 0 of
n
. Let us assume that the formula holds
for
n
, then the following chain of equations shows its validity for
n
=
1, by denoting
T
(
n
)
the corresponding sum. Of course,
T
(
0
)=
+
1:
n
(
n
+
1
)
2
(
n
+
1
)
=
(
n
+
1
)(
n
+
2
)
T
(
n
+
1
)=
+
(5.3)
2
2
2
(the reader is advised to search for a very beautiful geometrical proof of this formula,
due to Gauss, when he was a child).
