8.3 ARITHMETIC AND GEOMETRIC SEQUENCES
8.3 Airthmetic and Geometric Sequences
451
8.3 A R I T H M E T I C A N D G E O M E T R I C S E Q U E N C E S
Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I keep returning to the same problem. Why does the mathematics we have discovered in the past so often turn out to describe the workings of the Universe?
John Barrow
I remember that when I
was about twelve I learned
from [my uncle] that by the distributive law 1 times 1 equals 1. I thought that was great.
Peter Lax
Two kinds of regular sequences occur so often that they have specific names, arithmetic and geometric sequences. We treat them together because some obvious parallels between these kinds of sequences lead to similar formulas. This also makes it easier to learn and work with the formulas. The greatest value in this association is understanding how the ideas are related and how to derive the formulas from fundamental concepts. Anyone learning the formulas this way can recover them whenever needed.
Both arithmetic and geometric sequences begin with an arbitrary first term, and the sequences are generated by regularly adding the same number (the common difference in an arithmetic sequence) or multiplying by the same number (the common ratio in a geometric sequence). Definitions emphasize the parallel features, which examples will clarify.
Definition: arithmetic and geometric sequences
Arithmetic Sequence
a1 a and an an1 d for n 1
The sequence an is an arithmetic sequence with first term a and common difference d. Geometric Sequence
a1 a and an r ? an1 for n 1
The sequence an is a geometric sequence with first term a and common ratio r.
The definitions imply convenient formulas for the nth term of both kinds of sequences. For an arithmetic sequence we get the nth term by adding d to the first term n 1 times; for a geometric sequence, we multiply the first term by r, n 1 times.
Formulas for the nth terms of arithmetic and geometric sequences
For an arithmetic sequence, a formula for the nth term of the sequence is
an a n 1d.
(1)
For a geometric sequence, a formula for the nth term of the sequence is
an a ? r n1.
(2)
The definitions allow us to recognize both arithmetic and geometric sequences. In an arithmetic sequence the difference between successive terms, an1 an, is always the same, the constant d; in a geometric sequence the ratio of successive
terms, an1 , is always the same. an
452
Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers
Strategy: Calculate the differences and /or ratios of successive terms.
EXAMPLE 1 Arithmetic or geometric? The first three terms of a sequence are given. Determine if the sequence could be arithmetic or geometric. If it is an arithmetic sequence, find d; for a geometric sequence, find r.
(a) 2, 4, 8, . . .
(b) ln 2, ln 4, ln 8, . . .
(c)
1 ,
1,
1,
.
.
.
234
Solution
(a) a2 a1 4 2 2, and a3 a2 8 4 4. Since the differences are not the same, the sequence cannot be arithmetic. Checking ratios, a2 4 2, a1 2 and a3 8 2, so the sequence could be geometric, with a common ratio a2 4 r 2. Without a formula for the general term, we cannot say anything more
about the sequence.
(b)
a2
a1
ln
4
ln
2
ln
4 2
ln
2,
and
a3 a2 ln 8 ln 4
ln
8 4
ln
2,
so
the
sequence
could
be
arithmetic,
with
ln
2
as
the
common
difference. As in part (a), we cannot say more because no general term is given.
(c)
a2
a1
1 3
1 2
1 6
,
and
a3
a2
1 4
1 3
1 12
.
The
differences
are
not
the
same,
so
the
sequence
is
not
arithmetic.
a2 a1
1 3
1 2
2 ,
3
and
a3 a2
1 4
1 3
3 4
,
so
the
sequence
is
not
geometric.
Note
that
the
sequence
in
part
(a) could be geometric and the sequence in part (b) could be arithmetic, but
in part (c) you can conclude unequivocally that the sequence cannot be either
arithmetic or geometric.
EXAMPLE 2 Arithmetic or geometric? Determine whether the sequence is arithmetic, geometric, or neither.
(a) 3 1.6n (b) 2n (c) an ln n
Solution
(a) a2 a1 3 1.6 ? 2 3 1.6 ? 1 0.2 1.4 1.6, and a3 a2 3 1.6 ? 3 3 1.6 ? 2 1.6. From the first three terms, this could be an arithmetic sequence with d 1.6. Check the difference an1 an.
an1 an 3 1.6n 1 3 1.6n 1.6.
The sequence is arithmetic, with d 1.6. (b) a2 a1 4 2 2, and a3 a2 8 4 4, so the sequence is not
arithmetic. Using the formula for the general term,
an1 an
2n1 2n
2.
The sequence 2n is geometric, with 2 as the common ratio.
8.3 Airthmetic and Geometric Sequences
453
(c)
an1
an
lnn
1
ln
n
ln
n
n
1
.
The
difference
depends
on
n,
so
the sequence is not arithmetic. Checking ratios, an1 lnn 1 , so the ratio
an
ln n
also changes with n. The sequence is neither arithmetic nor geometric.
EXAMPLE 3 Arithmetic sequences Show that the sequence is arithmetic; find the common difference and the twentieth term.
(a) an 2n 1 (b) 50, 45, 40, . . . , 55 5n, . . .
Solution
(a) The first few terms of an are 1, 3, 5, 7, . . . , from which it is apparent that each term is 2 more than the preceding term; this is an arithmetic sequence with first term and common difference a 1 and d 2. Check to see that an1 an 2. To find a20, use either the defining formula for the sequence or Equation (1) for the nth term:
a20 2 ? 20 1 39 or a20 a 19d 1 19 ? 2 39.
(b) If bn 55 5n, then bn1 bn 55 5n 1 55 5n 5. This is an arithmetic sequence with a 50, d 5, and so b20 55 5 ? 20 45.
Given the structure of arithmetic and geometric sequences, any two terms completely determine the sequence. Using Equation (1) or (2), two terms of the sequence give us a pair of equations from which we can find the first term and either the common difference or common ratio, as illustrated in the next example.
EXAMPLE 4 Arithmetic sequences Suppose an is an arithmetic sequence with a8 6 and a12 4. Find a, d, and the three terms between a8
and a12.
Solution
From Equation (1), a8 a 7d, and a12 a 11d, from which the difference
is given by a12 a8 4d. Use the given values for a8 and a12 to get 4 6 4d,
or
d
5 2
.
Substitute
5 2
for
d
in
6
a
7d
and
solve
for
a,
a
47 2
.
Find
the
three
terms
between
a8
and
a12
by
successively
adding
5 2
:
a9
a8
5 2
7 2
,
a10
a9
5 2
1,
a11
a10
5 2
3. 2
Therefore,
a9
is
7 2
,
a10
is
1,
and
a11
is
3 2
.
EXAMPLE 5 Geometric sequences Determine whether the sequence is
geometric. If it is geometric, then find the common ratio and the terms a1, a3, and a10.
(a) 2n
(b)
2,
2 ,
2 ,
.
.
.
,
2
1
n1
,...
39
3
454
Chapter 8 Discrete Mathematics: Functions on the Set of Natural Numbers
Strategy: The property that identifies a geometric sequence is the common ratio: the values a2 , a3 , a4 , . . .
a1 a2 a3 must all be the same. For a geometric sequence, use Equation (2).
Solution
(a) The first few terms are 2, 4, 8, 16, . . . , each of which is twice the preceding
term. This is a geometric sequence with first term a 2, and common ratio
given
by
r
an1 an
2n1 2n
2.
Using
an
2n,
a1 2 a3 23 8 and a10 210 1024.
(b) Consider the ratio
2 1 n
an1
3
1 ,
an
2 1 n1
3
3
so
the
sequence
is
geometric
with
a
2
and
r
1 3
.
Using
an
2
1 3
n1,
we
get
a1
2,
a3
ar 2
2 9
,
and
a10
ar 9
2
1 3
9
2 19683
.
Partial Sums of Arithmetic Sequences
There is a charming story told about Carl Freidrich Gauss, one of the greatest mathematicians of all time. Early in Gauss' school career, the schoolmaster assigned the class the task of summing the first hundred positive integers, 1 2 3 ? ? ? 99 100. That should have occupied a good portion of the morning, but while other class members busied themselves at their slates calculating 1 2 3, 3 3 6, 6 4 10, and so on, Gauss sat quietly for a few moments, wrote a single number on his slate, and presented it to the teacher. Young Gauss observed that 1 and 100 add up to 101, as do the pair 2 and 99, 3 and 98, and so on up to 50 and 51. There are fifty such pairs, each with a sum of 101, for a total of 50 ? 101 5050, the number he wrote on his slate.
This approach works for the partial sum of any arithmetic sequence, and we will use the method to derive some useful formulas. However, the ideas are more valuable than memorizing formulas. If you understand the idea, you can recreate the formula when needed.
To find a formula for the nth partial sum of an arithmetic sequence, that is, the sum of n consecutive terms, pair the first and last terms, the second and next-to-last, and so on; each pair has the same sum. In fact, it is easier to pair all terms twice, as illustrated with Gauss' sum:
S100 1 2 ? ? ? 99 100 S100 100 99 ? ? ? 2 1 2S100 101 101 ? ? ? 101 101
The sum on the right has 100 terms, so 2S100 100101. Dividing by 2, S100 50101 5050.
For the general case, pairing the terms in Sn and adding gives 2Sn na1 an because there are n pairs, each with the same sum. Dividing by 2 yields the desired formula.
8.3 Airthmetic and Geometric Sequences
455
Partial sums of an arithmetic sequence
Suppose an is an arithmetic sequence. The sum Sn of the first n terms is given by
Sn
na1
2
an
(3)
The formula is probably most easily remembered as n times the average of the first and last terms.
Strategy: Let an 2n 1. To find S25 from Equation (3) requires a1 and a25, which the formula for an can provide. For (b), substitute 1 for a1 and 2n 1 for an in Equation (3) and simplify.
EXAMPLE 6 Partial sums For the sequence an 2n 1,
(a)
evaluate
the
sum
S25
25 k1
2k
1 and
(b) find a formula for Sn.
Solution Follow the strategy.
(a)
By
Equation
(3),
S25
25a1 2
a25 .
Now,
find
a1 and
a25 .
a1 2 ? 1 1 1 and a25 2 ? 25 1 49
Thus,
S25
251
2
49
625.
(b) In general,
Sn
na1
2
an
n1
2n 2
1
n2n 2
n 2.
Hence, Sn n2.
EXAMPLE 7 Arithmetic sequence The sum of the first eight terms of an arithmetic sequence an is 24; the sixth term is 0. Find a formula for an.
Solution For an, first find a and d. Since a6 a 5d, a 5d 0. Express S8 in terms of a and d,
S8
8a
a 2
7d
42a
7d.
Since we are given S8 24, Equation (3) states that 42a 7d 24. This gives a pair of equations to solve for a and d.
a 5d 0 2a 7d 6
We find d 2 and a 10. Therefore, the nth term is
an a n 1d 10 n 12 12 2n.
Partial Sums of Geometric Sequences
The idea of pairing terms, which works so well for arithmetic sequences, does not help with a geometric sequence. Another idea does make the sum easy to calculate
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