Download Sequences, part I

Sequences, part I
Mathematics, winter semester 2016/2017
October 5, 2016
Notation.
N denotes the set of all natural numbers, N = {1, 2, 3, . . .} .
Z denotes the set of all integers, Z = {0, 1, −1, 2, −2, 3, −3, . . .} .
Q denotes the set of all rational numbers, which are all numbers which may
be written as a fraction pq , where p, q ∈ Z and q 6= 0.
R denotes the set of all real numbers. The set of real numbers
consist of all
√
rational numbers as well as of irrational numbers
such
as
2
≈
1.41,
π ≈ 3.14.
Pn
For given real numbers a1 , a2 , . . . , an , i=1 ai denotes the sum a1 + a2 +
. . . + an .
For a ∈ R, |a| denotes the absolute value of the number a,
(
a
if a ≥ 0 √ 2
|a| =
= a .
−a if a < 0
√ 2
a must be non-negative
ATTENTION We always have ( a) = a (since
√
here), but sometimes (for negative as) a2 = −a.
Definitions
Definition. A sequence is a function a : N → R.
Definition. For n ∈ N the value a (n) is called the nth term of the sequence
a. The nth term (n ∈ N) of the sequence a is often denoted by an .
One may think about a sequence simply as about a list of real numbers (its
terms) arranged (written) in a specific order. To denote the sequence a we will
often write (an ) .
Definition. A sequence a is (strictly) increasing if for all n ∈ N, an+1 > an .
Definition. A sequence a is (strictly) decreasing if for all n ∈ N, an+1 < an .
Definition. A sequence a is non-decreasing if for all n ∈ N, an+1 ≥ an .
Definition. A sequence a is non-increasing if for all n ∈ N, an+1 ≤ an .
Definition. A sequence a is monotonic if it is increasing, decreasing, nondecreasing or non-increasing.
Definition. A sequence a is bounded if there exist (positive) real number
M such that for all n ∈ N, |an | ≤ M.
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Definition. A sequence a is called an arithmetic sequence if there exist
r ∈ R such that for all n ∈ N,
an+1 = an + r.
We have the following formula for the nth term of the arithmetic sequence a :
an = a1 + (n − 1) r.
We also have the following formula for the sum of the first n terms of the
arithmetic sequence a :
n
X
(n − 1) r
a1 + an
n = a1 +
n.
ai =
2
2
i=1
Definition. A sequence b is called a geometric sequence if there exist q ∈ R
such that for all n ∈ N,
bn+1 = bn q.
We have the following formula for the nth term of the geometric sequence b :
bn = b1 q n−1 .
We also have the following formula for the sum of the first n terms of the
geometric sequence b : if q 6= 1 then
n
X
bi = b1
i=1
1 − qn
.
1−q
Problems
1. Verify whether a sequence (an ) is (strictly) increasing or decreasing if
√
n
n
(a) an = n2 − n + 1; (b) an = 1−3
n + 1; (d) an = (−1) 2−n
2 ; (c) an =
n2 ;
2
1 2
−3
3+
)
(
n
(e) an = 3n+1
; (g) an = 2n − 6n + n1 .
1
n+3 ; (f) an =
n
2. A sequence (an ) is strictly increasing. What can be said about the monotonicity of the sequence (bn ) if (a) bn = 3an ; (b) bn = an + an+1 ; (c)
2
bn = (an ) ; (d) bn = an+1 − an ; (e) bn = (a )12 +1 .
n
3. Verify if the sequence an =
its terms satisfy
(−1)n
2n +4n
−
is monotonic, if it is bounded and if all
1
1
≤ an ≤ .
20
6
1
1
1
4. Let a1 = 11
20 and for n ≥ 2 let an = n+1 + n+2 + . . . + 2n . Verify if the
sequence (an ) is monotonic, if it is bounded and if all its terms satisfy
1
≤ an ≤ 1.
2
2
5. Find numbers x and y such that a1 = 32, a2 = x, a3 = y and a4 = 500
are first four terms of a geometric sequence. Then, compute the sum of
first seven terms of this sequence.
6. Prove that a sequence (an ) is a geometric sequence iff for all n ∈ N,
2
(an+1 ) = an · an+2 .
7. Find numbers x, y and z such that a1 = 1, a2 = x, a3 = y, a4 = z and
a5 = 257 are first five terms of an arithmetic sequence. Then, compute
the sum of first seventeen terms of this sequence.
8. Prove that a sequence (an ) is an arithmetic sequence iff for all n ∈ N,
2an+1 = an + an+2 .
9. Find a formula for the product of the first n terms of a geometric sequence.
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