CHAPTER 28. SEQUENCES: BASIC
(f) 1, 1, 2, 2, 3, 3, 4, 4, . . .
(g) 0, 1, 0, 1, 0, 1, . . .
28.4 Is the statement correct? Sequence {a
n
, n = 1, 2, 3, . . . } is equivalent to a function
f with domain of positive integers and f (n) = a
n
.
28.5 Find the 168th term of arithmetic sequence {a
n
, n = 1, 2, 3, . . . } with common
difference d = 1.5 and first term a
1
= 2.
28.6 Find the 8th term of geometric sequence {a
n
, n = 1, 2, 3, . . . } with common ratio
r = 0.5 and first term a
1
= 2.
28.7 The arithmetic mean of a and b is defined as
a +
b
2
. Prove that in an arithmetic
sequence, a term is the arithmetic mean of its two adjacent terms.
28.8 Given a > 0 and b > 0, their geometric mean is defined as
√
ab. Prove that in a
geometric sequence {a
n
} of only positive terms, a term is the geometric mean of its
two adjacent terms.
28.9 Arithmetic sequence {a
n
, n = 1, 2, 3, . . . } has common difference of 2. Compute the
values of a
3
− a
1
and a
100
− a
50
.
28.10 In arithmetic sequence {a
1
, a
2
, a
3
, . . . }, a
1
+ a
4
= 10. Find the value of a
2
+ a
3
.
28.11 In arithmetic sequence {a
1
, a
2
, a
3
, . . . }, a
9
+ a
11
= 10. Find the value of a
10
.
28.12 In arithmetic sequence {a
1
, a
2
, a
3
, . . . }, a
4
− a
1
= 12. Find its common difference.
28.13 In geometric sequence {a
1
, a
2
, a
3
, . . . },
a
4
a
1
= 10. Find its common ratio.
28.14 In geometric sequence {a
1
, a
2
, a
3
, . . . }, a
1
a
4
= 10. Find the value of a
2
a
3
.
28.15 Two arithmetic sequences {a
n
} and {b
n
} satisfy a
9
= b
9
and a
19
= b
19
. Prove a
n
= b
n
for all integer n ≥ 1.
28.16 Two geometric sequences {a
n
} and {b
n
} satisfy a
9
= b
9
and a
20
= b
20
. Prove a
n
= b
n
for any positive integer n.
28.17 Sequence {a
i
} is an arithmetic sequence. Suppose a
m
> a
n
for some integers m and
n satisfying m > n. Prove that the sequence is increasing.
28.18 If values of two terms of a geometric sequence are in interval [a, b], can any term
located between them in the sequence have a value outside [a, b]?
28.19 If terms a
3
and a
10
of a geometric sequence {a
n
, n = 1, 2, 3, . . . } are positive, can
any other term be negative?
c
Qishen Huang 87