Download Introductory Exercise

IB SL/HL 1.1
IB Diploma Mathematics SL/HL
Sequences and Series
Ms. Rola AbuSager
Please refer to the powerpoint presentation on Sequences and Series before reading the
lessons.
Lesson 1



Concept of sequences and series
Form of arithmetic sequences
nth term of arithmetic sequences
Notes
Title: Chapter 2: Sequences and Series
See p54
A sequence is an ordered list of numbers.
e.g.
A:
5, 10, 15, 20, …
B:
2, 6, 18, 54, …
C:
7, 4, 10, 3, …
Some sequences have a pattern; some do not. Sequences can be finite or infinite.
Each number in a sequence is call a term.
In sequence A, 5 is the first term, 10 is the second term etc. If there is a pattern, we can sometimes write an
expression for a general, nth term. The nth term in sequence A is 5n.
The letter u is used to represent a term. So, for sequence A,
u1 = 5
u2 = 10
un = 5n
A series is a sequence with the terms added together.
e.g.
5 + 10 + 15 + 20 + ….
Many sequences and series have a common structure. We will study two: arithmetic and geometric.
*****
Subtitle: Arithmetic Sequences
If the same number is added each time to get to the next term in a sequence then the sequence is arithmetic
(also called an arithmetic progression).
Sequence A above is an arithmetic sequence.
Such sequences are defined by the first term, a, and the common difference, d.
e.g.1.
e.g.2.
In A, u1 = 5 and d = 5
In the sequence 4, 7, 10, 13, … u1 = 4 and d = 3
For all arithmetic sequences we have the general formula
un = u1 + (n – 1) d
See examples 2-5 on p.57-58
Assignment #1: Pg. 58 #1a to 1e all and 2a to 2d
Due: Wednesday Sept. 30th, 2009
To: email address: [email protected] or [email protected]
Please email me if you have any questions!
IB SL/HL 1.1
Lesson 2
 Finite sum of an arithmetic series
Introductory Exercise
Calculate the sum of the integers from 1 to 100.
Notes
Title: The Finite Sum of an Arithmetic Series, Sn
See pg. 66-67
It can be shown that the sum of a finite number of terms of an arithmetic series is equal to
half the number of terms × (first term + last term)
Algebraically, the sum of the first n terms is
Sn =
n
(u1 + un)
2
or
n
( u1+ u1+ (n-1)d)
2
n
 Sn = (2u1 + (n-1)d)
2
Sn =
E.g. An arithmetic sequence has 5th term 7 and common difference 3. Find the sum of
the first 12 terms.
Method
Find u1
n
(2 u1 + (n-1)d)
2
Answer: 138
Use Sn =
Assignment #2: Pg. 67 Exercise 2E.2 #1a to 1d and 2a to 2c
Due: Thursday Oct. 1st. , 2009
To: email address: [email protected] or [email protected]
Please email me if you have any questions!
IB SL/HL 1.1
Lesson 3
 Sigma notation (not limited to arithmetic series or i = 1 to ….)
Another way to express a sum is to use sigma notation.

(sigma) is the first letter of the Greek word for ‘sum’
100
E.g.1 1 + 2 + 3 + 4 + …. + 100 can be written as  i
i 1
Read as, “the sum as i goes from 1 to 100 of i”.
E.g.2 6 + 11 + 16 + 21 + ….. + 36 can be written as
7
 (1  5n)
n 1
The “counting” letter is called the index. i, n and k are commonly used. The index does
6
not have to start at 1. The series in example 2 can also be written as  (6  5n) , for
n 0
example.
Of course, sigma notation can be used to represent other kinds of series, not just
arithmetic.
Example
Write the series below using sigma notation.
2 + 5 + 10 + 17 + …. + 82
9
Answer:
 (n
2
 1)
n 1
Assignment #3: Pg. 68 #3.7.11. & 13
Due: Saturday Oct. 3rd. , 2009
To: email address: [email protected] or [email protected]
Please email me if you have any questions!
Good Luck!