ARITHMETIC SEQUENCES AND SERIES Week Commencing Monday 28 th
September Learning Intention: To be able to find the nth term of an
arithmetic sequence or series. To be able to find the number of
terms in an arithmetic sequence or series. Contents: 1.What is an
Arithmetic Sequence?What is an Arithmetic Sequence? 2.What is an
Arithmetic Series?What is an Arithmetic Series? 3.Assignment
2Assignment 2 4.Finding terms of Arithmetic Sequences and
SeriesFinding terms of Arithmetic Sequences and Series 5.Number of
terms in a Sequence or SeriesNumber of terms in a Sequence or
Series 6.Finding first term and common differenceFinding first term
and common difference 7.Assignment 3Assignment 3
Slide 2
ARITHMETIC SEQUENCES AND SERIES What is an Arithmetic Sequence?
An arithmetic sequence is a sequence that increases by a constant
amount each time. It can be defined by the recurrence relationship:
U n+1 = U n + k, where k is a constant number Examples of
arithmetic sequences are: 5, 8, 11, 14, 17,... increasing by 3 each
time 100, 95, 90, 85,... increasing by -5 each time
Slide 3
ARITHMETIC SEQUENCES AND SERIES What is an Arithmetic Series?
If you add together the terms of an arithmetic sequence we get an
arithmetic series the same terms but instead of commas separating
them it is a + sign. Examples of arithmetic series are: 5 + 8 + 11
+ 14 + 5 + 1 + -3 + -7 + -11 +
Slide 4
ARITHMETIC SEQUENCES AND SERIES Assignment 2 What are
Arithmetic Sequences & Series? Follow the link for Assignment 2
on Arithmetic Sequences and Series in the Moodle Course Area.
Completed assignments must be submitted by 5:00pm on Monday 5 th
October.
Slide 5
ARITHMETIC SEQUENCES AND SERIES Finding terms of Arithmetic
Sequences and Series For both arithmetic sequences and series the
first term is generally called a and the constant it increases by
is called the common difference, d. We can use a and d to help us
find the nth term of an arithmetic sequence or series. The formula
for the nth term is given by: a + (n 1)d where n is term we are
looking for a is the first term d is the common difference
Slide 6
ARITHMETIC SEQUENCES AND SERIES Terms of an Arithmetic Series
Example: Find the 10 th, 20 th and nth terms of this arithmetic
series: + 1 + 1 + 2 + Solution: a = d = 1 = Using a + (n -1)d (i)10
th term = + (10 1) = + (9) = 7 a = d = 1 = Using a + (n -1)d (ii)
20 th term = + (20 1) = + (19) = 14 a = d = 1 = Again, using a + (n
-1)d (iii) n th term = + (n 1) = + n = n
Slide 7
ARITHMETIC SEQUENCES AND SERIES Number of Terms in a Series If
we know the final term in a sequence or series we can use a and d
to help us find how many terms there are in sequence or
series.
Slide 8
ARITHMETIC SEQUENCES AND SERIES Number of Terms in a Series?
Example: How many terms are in this arithmetic series: 0.7 + 0.3 +
-0.1 + -0.5 + + -5.7 Solution: We know the last term is -5.7, a =
0.7 and d = -0.4. We can therefore use the formula a + (n 1)d to
form an equation and solve for n. We get: 0.7 + ( n 1)(-0.4) = -5.7
0.7 0.4n + 0.4 = -5.7 (multiplying out brackets) -0.4n = -6.8
(taking numbers to one side) n = -6.8 / -0.4 = 17 (dividing by
0.4)
Slide 9
ARITHMETIC SEQUENCES AND SERIES Finding a and d A very popular
type of question to be asked in the exam is to find the first term
and the common difference when given what two of the terms in the
series are.
Slide 10
ARITHMETIC SEQUENCES AND SERIES Finding a and d Example: The
seventh term in an arithmetic series is 15 and the eight term is
20. Find the first term. Solution: U 7 = 15 and U 8 = 20, therefore
d = 5. Furthermore: a + (7 -1)(5) = 15 a + 30 = 15 a = 15 30 =
-15
Slide 11
ARITHMETIC SEQUENCES AND SERIES Finding a and d Example: Given
that the 3 rd term of an arithmetic series is 30 and the 10 th term
is 9 find a and d. Hence find which term if the first one to become
negative. Solution: U 3 = 30 and U 10 = 9 a + (3 -1)d = 30a + (10
1)d = 9 a + 2d = 30(1)a + 9d = 9(2) We solve equations (1) and (2)
simultaneously to find a and d. Subtracting (1) from (2) gives: 7d
= -21 d = -3 Therefore, a + 2(-3) = 30 a 6 = 30 a = 36 a = 36d = -3
We want the first term to become negative i.e a + (n 1)d < 0
Using the a and d we have found we get: 36 + (n 1)(-3) < 0 36 3n
+ 3 < 0 -3n < -39 n > 13 That is, from term number 14
onwards the number will be negative.
Slide 12
ARITHMETIC SEQUENCES AND SERIES Assignment 3 Finding terms of
an Arithmetic Series. Follow the link for Assignment 3 on Finding
terms of an Arithmetic Series in the Moodle Course Area. This is a
Yacapaca Activity. Completed assignments must be submitted by
5:00pm on Monday 5 th October.