Sequences and Series (2) Generate a sequence using inductive

1
Sequences and Series (2)

• Generate a sequence using inductive definition
• Describing the behaviour of a sequence

2
Inductive Definition
Write down the terms of the sequence un 5n - 2
u1 3 u2 8 u3 13 u4 18
3
Inductive Definition
u1 3 u2 8 u3 13 u4 18
Each term is 5 more than the last
u2 u1 5
We could write this as
u3 u2 5
un1 un 5
u4 u3 5
Next term
Current term
4
Inductive Definition
un1 un 5
This is called a recurrence relation
5
Inductive Definition
un1 un - 3
Write down a sequence using the recurrence
relation above.
What extra information do I need?
6
Inductive Definition
un1 un - 3
In order to write a sequence when given a
recurrence relation, I need to know the first
term.
7
Inductive Definition
un1 un - 3
u1 7
Now write the sequence
u1 7 u2 4 u3 1 u4 -2 u5 -5
When we are given BOTH the first term and the
recurrence relation we have an Inductive
definition.
8
Example 1
The sequence 2, 10, 34, 106 is defined by un1
aun b where a and b are constants
Find the value of a and b.
1
u2 au1 b
10 2a b
2
u3 au2 b
34 10a b
2-1
Simultaneous Equations!!!
24 8a
a 3 b 4
un1 3un 4
9
Example 2
A sequence is defined by u1 3 un1 un k
If u4 7.5, find k
u1 3
u2 3 k
u3 3 k k 3 2k
u4 3 2k k 3 3k
3 3k 7.5
k 1.5
Un1 un 1.5
10
Activity
Turn to page 33 of your textbook and answer
questions in exercise B.