Transcript What are Arithmetic Sequences & Series?

ARITHMETIC SEQUENCES
AND SERIES
Week Commencing Monday 28th 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?
2. What is an Arithmetic Series?
3. Assignment 2
4. Finding terms of Arithmetic Sequences and
Series
5. Number of terms in a Sequence or Series
6. Finding first term and common difference
7. Assignment 3
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:
Un+1 = Un + 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
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 comma’s separating them it is
a “+” sign.
Examples of arithmetic series are:
5 + 8 + 11 + 14 + …
5 + 1 + -3 + -7 + -11 + …
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 5th October.
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
ARITHMETIC SEQUENCES
AND SERIES
Terms of an Arithmetic Series
Example:
Find the 10th, 20th and nth terms of this
arithmetic series:
¼ + 1 + 1¾ + 2½ + …
Solution:
a = ¼
d = 1 – ¼ = ¾
Using ausing
Again,
+ (n a-1)d
+ (n -1)d
th
thterm
(i)
(iii)
(ii) 10
20
nth
term
term = ¼ + (n
(10
(20– –1)¾
1)¾
= ¼ + ¾n
(9)¾
(19)¾
– ¾
= ¾n
7
14½– ½
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.
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)
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.
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:
U7 = 15 and U8 = 20, therefore d = 5.
Furthermore:
a + (7 -1)(5) = 15
a + 30 = 15
a = 15 – 30 = -15
ARITHMETIC SEQUENCES
AND SERIES
Finding a and d
Example:
Given that the 3rd term of an arithmetic series is
30 and the 10th term is 9 find a and d. Hence
find which term if the first one to become
negative.
Solution:
aU= =36
30
d = -3
and
U10 = 9
We
want
first term to become
negative
a + (3
-1)dthe
= 30
a + (10
– 1)d =i.e
9
3
aa ++ (n
2d–=1)d
30 < 0 (1)
a + 9d = 9
(2)
Using
the equations
a and d we(1)
have
we get:
We solve
andfound
(2) simultaneously
to find
a and
36
+ (nd.– 1)(-3) < 0
Subtracting
36
– 3n + 3 < (1)
0 from (2) gives:
7d =< -21
-3n
-39
-3
nd >= 13
That is, from term number 14 onwards the number
Therefore,
will
be negative.a + 2(-3) = 30
a – 6 = 30
a = 36
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 5th October.