Download Chapter 2 Sequences and Series

CHAPTER 2 SEQUENCES
AND SERIES
Tori Beals, 3rd Hour
Trevor Campbell, 5th Hour
NUMBER PATTERNS (SECTION A)
A list of numbers where there is a pattern is
called a number sequence
 The numbers in the sequence are said to be its
members or terms

Example of a number sequence 2, 4, 6, 8, 10, 12
 The third term of this sequence is 6

SEQUENCES OF NUMBERS
(SECTION B)
A number sequence is a set of numbers defined
by a rule that is valid for positive integers
 The general term or nth term is represented by
Un
 Sequences may be defined in one of the following
ways

Listing the first few terms and assuming that the
pattern represented continues indefinitely
 Giving a description in words
 Using a formula which represents the general term
or nth term

CONT.

Example, Section 2B, #1). List the first five terms
of:
A) {2n}
2, 4, 6, 8, 10
B) {2n+2}
4, 6, 8, 10, 12
C) {2n-1}
1, 3, 5, 7, 9
D) {2n-3}
-1, 1, 3, 5, 7
ARITHMETIC SEQUENCES
(SECTION C)
An arithmetic sequence is a sequence in which
each term differs from the previous one by a fixed
number
 Un is arithmetic if and only if U(n+1)=d for all
positive n integers where d is a constant called
the common difference
 For an arithmetic sequence with the first term
U1 and common difference d the general term or
nth term is
Un=U1+(n-1)d (The general term formula)

CONT.

The name ‘arithmetic’:
If a, b, and c are any consecutive terms of an
arithmetic sequence then:
b-a= c-b
2b= a+c
b=(a+c)/2
So, the middle term is the arithmetic mean
of the terms on either side of it.
CONT.

Example 2C. 5) A
sequence is defined by
Un=3n-2
 A)
Prove the sequence is
arithmetic (hint: find
U(n+1)-Un)
If n= 2 then
U2=3(2)-2
U2=4
and
U1=3(1)-2
U1=1

D=3

If n=4 then
U5=3(5)-2
U5=13
And
U4=3(4)-2
U4=10
D=3
CONT.

5c. Find the 57th term.
Un = 3n −2
U57 = 3(57) – 2
U57 = 171 −2
U57 = 169
5d. What is the least
term of the sequence
which is greater than
450?
U150 = 448
U151 = 451
U152 = 454
The 151st term is the
first (least) term of the
sequence that is
greater than 450.

GEOMETRIC SEQUENCES
(SECTION D)

A sequence is geometric if each term can be
obtained from the previous one by multiplying
the same non-zero constant.
For example: 2, 10, 50, 250, … is a geometric
sequence.
2×5= 10, 10×5= 50, 50×5= 250
Each term divided by the previous one gives the same
constant.

o
The general term for a geometric sequence is:
Un = U1 rn−1
CONT.
The name ‘geometric’:
 If a, b, and c are consecutive integers of a
geometric sequence, then b/a = c/b.
 Compound Interest:
General Compound Interest formula:
Un+1 = U1 × rn

CONT.

Example, Section 2D, #4:
a. Show that the sequence 5, 10, 20, 40, … is geometric.
10/5= 2, 20/10= 2, 40/20= 2. Therefore, U1 = 5, and r = 2.


Find Un and hence find the 15th term.
Un = (5)(2)n-1
U15 = (5)(2)15−1
(5)(2)14
(5)(16,384)
U15 = 81,920
CONT.

Example, Section 2D, #4:
a. Show that the sequence 5, 10, 20, 40, … is geometric.
10/5= 2, 20/10= 2, 40/20= 2. Therefore, U1 = 5, and r = 2.


Find Un and hence find the 15th term.
Un = (5)(2)n-1
U15 = (5)(2)15−1
(5)(2)14
(5)(16,384)
U15 = 81,920
SERIES
(SECTION E)
A series is the addition of the terms of a
sequence.
 For the series { Un }, the corresponding series is
 U1 + U2 + U3 + … + Un
 The sum of a series is the result when we
perform the addition.
 Given a series which includes the first n terms of
a sequence, its sum is Sn = U1 + U2 + U3 + … Un .

SIGMA NOTATION
Series can be written more compactly using
sigma notation.
uk
 U1 + U2 + U3 + … Un is written as
 This reads “the sum of all numbers of the form uk
where k = 1, 2, 3, … , up to n”.

PROPERTIES OF SIGMA NOTATION
CONT.
An arithmetic series is the addition of
successive terms of an arithmetic sequence.
 Sum of an arithmetic series:
Sn = n/2( u1 + un)
OR
Sn = n/2(2u1 + (n −1)d)

CONT.
A geometric series is the addition of successive
terms of a geometric sequence.
 Sum of a geometric series:
Sn = u1 (rn – 1) ÷ (r −1)
OR
Sn = u1 (1 − rn) ÷ (1 − r)

SUM OF AN INFINITE GEOMETRIC SERIES
When n becomes very large, Sn will change.
 If r > 1, the series is divergent and the sum
becomes infinitely large.
 If -1 < r < 1, then rn approaches 0 for very large n.
 We say the series converges. Its sum is written
as:
Sn = u1 ÷ (1 –r), for r < 1

MORE EXAMPLE PROBLEMS
Section 2C, #7a. :
Find k given the consecutive arithmetic terms.
32, k, 3
k – 32 = 3 – k
2k – 32 = 3
2k = 35
k = 17.5

MORE EXAMPLE PROBLEMS (CONT.)
Section 2D, Example 10, page 79.
 $5000 is invested for 4 years at 7% p.a. compound
interest, compounded annually. What will it
amount to at the end of this period?
5000 × (1.07)4
≈ 6553.98
It amounts to about $6553.98

MORE EXAMPLE PROBLEMS (CONT.)
Section 2E.1, #2a.
Expand and evaluate:

( Expand 3k – 5, starting with the first term of -2 and
stopping at the fourth term, 7).
= -(2) + 1 + 4 + 7
= 10
MORE EXAMPLE PROBLEMS (CONT.)
Section 2E.2, #1a.
 Find the sum of:
3 + 7 + 11 + 15 + … to 20 terms.
U1 = 3 d = 4 n = 20
S20 = 20/2 (2(3) + (20 −1)4)
10(6 + 76)
10(82)
S20 = 820

MORE EXAMPLE PROBLEMS (CONT.)
Section 2E.3, #1a.
 Find the sum of the following series:
12 + 6 + 3 + 1.5 + … to 10 terms.
U1 = 12 r = ½ n = 10
S10 = (12( 1 – 1/2 10) ÷ (1 – ½ )
(12(1 −1/1024) ÷ ½
(12(1023/1024)) ÷ ½
(11.99) ÷ ½
S10 ≈ 23.98
S10 ≈ 24

GOLDEN PACKET MATERIALS
General term of an arithmetic sequence
 General term of a geometric sequence
 Sum of the terms of an arithmetic sequence
 Sum of the terms of a finite geometric sequence
 Sum of an infinite geometric sequence

NEED TO MEMORIZE

Compound Interest Formula