Arithmetic and Geometric Sequences
Felix Lazebnik
This collection of problems
1
is for those who wish to learn about arithmetic
and geometric sequences, or to those who wish to improve their understanding
of these topics, and practice with related problems. The attempt was made to
illustrate some ties between these and other notions of mathematics. The prob
lems are divided (by horizontal lines) into three groups according to di
ﬃ
culty:
easier, average, and harder.
Of course, the division is very subjective.
The
collection is aimed to freshmen and sophomores college students, to good high
school students, and to their teachers.
∗ ∗ ∗
An infinite sequence of numbers
{
a
n
}
n
≥
1
is called an
arithmetic
sequence,
if there exists an number
d
, such that for every
n
≥
1,
a
n
+1
=
a
n
+
d
:
∃
d
∀
n
∈
N
(
a
n
+1
=
a
n
+
d
)
.
The number
d
is called the
common di
ff
erence
, or just the
di
ff
erence
of the
arithmetic sequence.
Examples:
a
1
= 1
, d
= 1 :
1
,
2
,
3
,
4
,
5
, . . .
;
a
1
= 12
, d
= 8
.
5 :
12
,
20
.
5
,
29
,
37
.
5
, . . .
;
a
1
= 5
, d
=
−
1 :
5
,
1
,
−
3
,
−
7
, . . .
;
a
1
= 2
, d
= 0 :
2
,
2
,
2
,
2
, . . .
.
An infinite sequence of numbers
{
b
n
}
n
≥
1
is called a
geometric
sequence, if
there exists an number
r
, such that for every
n
≥
1,
b
n
+1
=
b
n
r
:
∃
r
∀
n
∈
N
(
b
n
+1
=
b
n
r
)
.
The number
r
is called the
common ratio
, or just the
ratio
of the geometric
sequence.
Examples:
b
1
= 2
, r
= 1 :
2
,
2
,
2
,
2
, . . .
;
b
1
= 1
, r
= 2 :
1
,
2
,
4
,
8
, . . .
;
b
1
= 3
, r
= 5 :
3
,
15
,
75
,
375
, . . .
;
b
1
=
−
3
, r
=
−
1
/
2 :
−
3
,
3
/
2
,
−
3
/
4
,
3
/
8
, . . .
.
1
Last revision: December 20, 2008.
1
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Often an arithmetic (a geometric) sequence is called an arithmetic (a geo
metric)
progression
. The term comes from Latin
progredior
– ‘walk forward’;
progressio
– ‘movement forward’, ‘success’. Problems about progressions go back
to
Rhind Papyrus
, c. 1550 BC and Babylonian astronomical tables, c. 25002000
BC.
1. Let
{
a
n
}
n
≥
1
be an arithmetic sequence with di
ff
erence
d
. Prove that
(a)
a
n
=
a
1
+ (
n
−
1)
d,
for all
n
≥
2
(b)
a
1
+
a
n
=
a
2
+
a
n
−
1
=
a
3
+
a
n
−
2
=
. . .
(c)
S
n
=
a
1
+
a
2
+
· · ·
+
a
n
=
a
1
+
a
n
2
n
=
2
a
1
+(
n
−
1)
d
2
n
for all
n
≥
2 (and
for
n
= 1, if we assume
S
1
=
a
1
).
2. Let
{
b
n
}
n
≥
1
be a geometric sequence with ratio
r
. Prove that
(a)
b
n
=
b
1
r
n
−
1
, for all
n
≥
2
(b)
b
1
b
n
=
b
2
b
n
−
1
=
b
3
b
n
−
2
=
. . .
(c) Let
n
≥
2, and
S
n
=
b
1
+
b
2
+
· · ·
+
b
n
. Then
S
n
=
n
i
=1
b
i
=
b
1
−
b
n
r
1
−
r
=
b
1
(1
−
r
n
)
1
−
r
,
if
r
= 1
nb
1
,
if
r
= 1
.
The formula also holds for
n
= 1, if we assume
S
1
=
b
1
.
(d) If

r

<
1, then lim
n
→∞
r
n
= 0, and
∞
i
=1
b
i
=
b
1
1
−
r
.
3. Given that each sum below is the sum of a part of an arithmetic or geo
metric progression, find (or simplify) each sum.
(a) 75 + 71 + 67 +
· · ·
+ (
−
61).
(b) 75 + 15 + 3 +
...
+
3
5
7
.
(c) 1 + 2 + 3 +
. . .
+ (
n
−
1) +
n
.
(d)
i
+ (
i
+ 1) + (
i
+ 2) +
· · ·
+
j
, where
i, j
∈
Z
,
i < j
.
(e) 1 + 3 + 5 +
· · ·
+ (2
n
−
1), where
n
∈
N
.
(f)
x
i
+
x
i
+1
+
x
i
+2
+
· · ·
+
x
j
, where
i, j
∈
Z
,
i < j
,
x
= 1.
4. Suppose that the sum of the first
n
terms of an arithmetic sequence is
given by the formula
S
n
= 4
n
2
−
3
n
for every
n
≥
1.
Find three first
terms of the arithmetic sequence and its di
ff
erence.
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 Fall '10
 CIOABA
 Arithmetic progression, Prime number, Geometric progression

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