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∑
i
2
∑
=
nT
n
−
T
i
(5.5)
i
=
1
,
n
i
=
1
,
n
−
1
but by formula 5.4 we have:
2
i
∑
T
i
=
∑
i
∑
i
2
+
∑
i
i
(
i
+
1
)
/
2
=
1
/
i
=
1
,
n
=
1
,
n
i
=
1
,
n
=
1
,
n
that is:
∑
i
=
1
,
n
i
2
∑
i
=
1
,
n
T
i
−
∑
i
=
1
,
n
=
2
i
(5.6)
therefore, by equating the right members of Eqs. (5.5) and (5.6), we have:
∑
∑
T
i
−
∑
i
nT
n
−
T
i
=
2
i
i
=
1
,
n
−
1
i
=
1
,
n
=
1
,
n
that is:
∑
∑
nT
n
−
T
i
=
2
T
i
+
2
T
n
−
T
n
i
=
1
,
n
−
1
i
=
1
,
n
−
1
∑
i
=
1
,
n
−
1
3
T
i
=(
n
−
1
)
T
n
T
i
=
(
n
−
1
)
T
n
∑
i
=
1
,
n
−
1
.
3
Finally, from Eq. (5.5) it follows:
nT
n
−
(
n
−
1
)
T
n
∑
i
2
=
3
i
=
1
,
n
3
nT
n
−
(
n
−
1
)
T
n
=
3
T
n
(
3
n
−
n
+
1
)
=
3
n
(
n
+
1
)(
2
n
+
1
)
=
.
6
Formulae for the sums of powers (up to the 17th power) were discovered by Faul-
haber, a mathematician of the 17th century. Faulhaber's formulae follow a general
pattern which can be proved by induction.
>
Theorem 5.3 (Faulhaber).
The sum of powers of degree k, for k
0
, has the fol-
lowing general form, where P
k
(
n
)
is a polynomial of degree k:
n
i
=
1
i
k
n
k
+
1
k
=
1
+
P
k
(
n
)
+
