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S
k
(
k
+
n
nS
k
−
1
(
n
−
1
)
−
Q
k
(
n
−
1
)=
S
k
(
n
−
1
)
which, adding
n
k
to the both members, becomes:
S
k
(
k
+
n
n
k
n
k
nS
k
−
1
(
n
−
1
)
−
Q
k
(
n
−
1
)+
=
S
k
(
n
−
1
)+
that is
+
k
1
n
k
S
k
(
n
)=
nS
k
−
1
(
n
−
1
)
−
Q
k
(
n
−
1
)+
k
nS
k
−
1
(
n
k
k
S
k
(
n
)=
n
−
1
)
−
Q
k
(
n
−
1
)+
k
+
1
and, by Eq. (5.7):
n
n
k
k
k
S
k
(
n
)=
[(
n
−
1
)
/
k
+
P
k
−
1
(
n
−
1
)]
−
Q
k
(
n
−
1
)+
k
+
1
and finally:
n
kn
k
1
k
S
k
(
n
)=
(
n
−
1
)
+
kP
k
−
1
(
n
−
1
)
−
kQ
k
(
n
−
1
)+
k
+
1
which corresponds to the statement of the theorem.
5.6.3
Geometrical Progressions
A second case of growth is that of a population where at each step each individual
generates
k
offspring, therefore at each step the population size increases of its size
multiplied by
k
(no external effect and no death is considered). In this case, assuming
an initial size equal to 1, the formula providing the population size at generation step
n
is:
n
i
=
0
k
i
G
(
n
)=
.
We prove that
k
n
−
1
1
.
In fact, the proof of this equation is the following:
G
(
n
)=
k
−
n
+
1
i
=
1
k
i
n
i
=
0
k
i
k
n
+
1
kG
(
n
)
−
G
(
n
)=
−
=
−
1
that is,
k
n
+
1
G
(
n
)(
k
−
1
)=
−
1
