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Homework Questions. Number Patterns. Find the next two terms, state a rule to describe the pattern. 1, 3, 5, 7, 9… 16, 32, 64… 50, 45, 40, 35… -3, -7, -11, -15…. Sequence Notation. A sequence is an ordered list of numbers – each number is a term . State the first 5 terms: a n = n - PowerPoint PPT Presentation
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Homework Questions
Number Patterns
Find the next two terms, state a rule to describe the pattern.
1. 1, 3, 5, 7, 9…2. 16, 32, 64…3. 50, 45, 40, 35…4. -3, -7, -11, -15…
Sequence Notation
A sequence is an ordered list of numbers – each number is a term.
State the first 5 terms: an = n (plug in 1, 2, 3, 4, 5) 1, 2, 3, 4, 5
More Examples
1. an = 4n
2. an = 2n-3
3. an = |1-n2|
4. an =
5. an = 3
1
n
3
6n
Recursive v. Explicit
Definition Recursive Formula – a sequence is
recursively defined if the first term is given and there is a method of determining the nth tem by using the terms that precede it.
English – if you can use the term before it to figure out what comes next
Ex: {-7, -4, -1, 2, 5, …}
Examples of Recursive
{-9, -4, -2, 0, 2, …}
{-4, -8, -16, -32, -64, …}
{6, 11, 16, 21, 26, …}
{8, 4, 2, 1, …}
Definition
Explicit Formula – a formula that allows direct computation for any term for a sequence
English – you don’t need to term prior in order to figure out what the nth term is going to be.
Ex: {8, 9, 10, 11, 12, …} an= n + 7
Examples of Explicit
{-3, 1, 5, 9, …}
{1, 4, 9, 16, …}
{7, 9, 11, 13, …}
{24, 20, 16, 12, …}
Arithmetic Sequences
Arithmetic Sequences
In an arithmetic sequence, the difference between consecutive terms is constant.
The difference is called the common difference.
To find d: 2nd term – 1st term
Arithmetic?
1. 2, 4, 8, 16
2. 6, 12, 18
3. 48, 45, 42
4. 2, 5, 7, 12
Arithmetic Sequence Formulas Recursive
Formula an = an-1 + d use if you know
prior terms Explicit Formula
an = a1 + (n-1)d
an = nth term a1 = 1st term n = number of
terms d = common
difference
Examples
Find the 20th term of each sequence
1. 213, 201, 189, 177…
2. .0023, .0025, .0027…
More examples
Find the 17th term of the sequence:
3. a16 = 18, d = 5
Find the missing term
Use arithmetic mean = average!4. 84, _______, 110
5. 24, _______, 57
Geometric Sequences
Geometric Sequences
In a geometric sequence, the ratio between consecutive terms is constant.
This ratio is called the common ratio.
To find r: stterm
ndterm
1
2
Geometric, Arithmetic, Neither?(find the next 2 terms if so)
1. 5, 15, 45, 135…
2. 15, 30, 45, 60…
3. 6, -24, 96, -384…
4. 8, 20, 32, 44…
Geometric Sequences Formulas
Recursive Formula an = an-1 r
Explicit Formula an = a1 rn-1
Find the 19th term…
1. 11, 33, 99, 297…
2. 20, 17, 14, 11, 8…
FYI - Graphs
Arithmetic Graphs are linear
Geometric Graphs are exponential
Geometric Mean
Geometric Mean =
3. 20, _____, 80
4. 3, ____, 18, 75
5. 28, ____, 5103
numbersproductof 2
Homework
WORKSHEET! We need to talk about numbers
16-20 though, so wait on me!