
Chapter 2 6
Finite Sums and Infinite S eries
26.1 Finite Numerical Sums
26.1.1 Progressions
Arithmetic progression:
1.
n−1
X
k=0
(a + bk) = an +
bn(n −1)
2
.
Geometric progression:
2.
n
X
k=1
aq
k−1
= a
q
n
− 1
q − 1
.
Arithmetic-geometric progression:
3.
n−1
X
k=0
(a + bk)q
k
=
a(1 − q
n
) − b(n − 1)q
n
1 −q
+
bq(1 −q
n−1
)
(1 − q)
2
.
26.1.2 Sums of Powers of Natural Numbers Having the Form
P
k
m
1.
n
X
k=1
k =
n(n + 1)
2
.
2.
n
X
k=1
k
2
=
1
6
n(n + 1)(2n + 1).
3.
n
X
k=1
k
3
=
1
4
n
2
(n + 1)
2
.
4.
n
X
k=1
k
4
=
1
30
n(n + 1)(2n + 1)(3n
2
+ 3n −1).
5.
n
X
k=1
k
5
=
1
12
n
2
(n + 1)
2
(2n
2
+ 2n − 1).
6.
n
X
k=1
k
m
=
n
m+1
m + 1
+
n
m
2
+
1
2
C
1
m
B
2
n
m−1
+
1
4
C
3
m
B
4
n
m−3
+
1
6
C
5
m
B
6
n
m−5
+ ···.
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