activity – sets of 3 r th term r+1 th term first few terms
TRANSCRIPT
Activity – sets of 3Activity – sets of 3
rrthth term term
r+1r+1thth term term
first few termsfirst few terms
Sequences and SeriesSequences and Series Use subscript notationUse subscript notation
Generate a sequence using a ruleGenerate a sequence using a rule
Generate a sequence using inductive definitionGenerate a sequence using inductive definition
Describing the behaviour of a sequenceDescribing the behaviour of a sequence
SequencesSequences
Finite Infinite
2 4 3 1 4 1, 4, 7, 10…
A finite sequence has a first and last term
A infinite sequence goes on forever
NotationNotation
If we say un = 3n - 5
u1 = 3(1) - 5
u2 = 3(2) - 5
u3 = 3(3) - 5
u4 = 3(4) - 5
u5 = 3(5) - 5
= -2
= 1
= 4
= 7
= 10
SequencesSequences
For each of the following, work out the first 6 terms, the 50th term and the 100th term. Describe what happens as n gets larger.
un = 3n + 2
cn = 7 – 4n
kn = n2 – 2n + 8
rn = (-2)n
ntn
12
Inductive DefinitionInductive Definition
u1 = 3 u2 = 8 u3 = 13 u4 = 18
Write down the terms of the sequence un = 5n - 2
Inductive DefinitionInductive Definition
u1 = 3 u2 = 8 u3 = 13 u4 = 18
Each term is 5 more than the last
u2 = u1 + 5
u3 = u2 + 5
u4 = u3 + 5
We could write this as:
un+1 = un + 5
Next term Current term
Inductive DefinitionInductive Definition
This is called a recurrence relation
un+1 = un + 5
Inductive DefinitionInductive Definition
Write down a sequence using the recurrence relation above.
un+1 = un - 3
What extra information do I need?
Inductive DefinitionInductive Definition
In order to write a sequence when given a recurrence relation, I need to know the first term.
un+1 = un - 3
Inductive DefinitionInductive Definition
Now write the sequence
un+1 = un - 3u1 = 7
u1 = 7 u2 = 4 u3 = 1 u4 = -2 u5 = -5
ProblemProblemA sequence is defined by u1 = 3 un+1 = 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
Un+1 = un + 1.5
QuizQuiz