Transcript Sequences and Functions

Patterns and Sequences
Classification of Sequences
Identification of Essential Determiners
What are we doing?
All in the name of
education, Watson!!!
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We will determine if patterns are sequences.
If so……what kind are they?
What are their determining features?
Do those features have a specific name?
• So…..this is really about naming things we know about, using
names we haven’t used before!
Vocabulary
• A sequence is
a set of
numbers
that follow a pattern.
Vocab
moment…
alter the accent.
• A term is
a number within a sequence.
a series of numbers whose terms
• An arithmetic sequence is
are found by adding a fixed value to a previous term, + or -.
the fixed number that is added to
• A common difference is
(combined with) the previous term in a sequence.
a series of numbers whose terms
• A geometric sequence is
are found by multiplying a fixed value to a term.
the fixed value multiplied with the
• A common ratio is
previous term. . Common multiplier, scale factor….
Examples…….
A swimmer training for a meet swims 5 laps the first day, 6 ½ laps the next
day, 8 laps the third day, and so on. Find the next three terms of the sequence.
Write a rule to describe the sequence.
1st term to 19th? 1 + _____ = 19
5
6½
+1½
8
+1½
9½
We will use the common difference
of 1 ½ to find the next three terms.
+1½
11
+1½
You can see the next three terms are 9 ½ , 11, and 12 ½ .
12 ½
+1½
The rule, as we have stated in previous lessons, would be to Start with 5 and add
1 ½ repeatedly.
What would the 19th term be??????? How could this rule be
expressed as an equation?
5 + 1 ½ n= 5 + 1 ½ (18)=
5 + 27 = 32
Find the Common Ratio
Find the common ratio in the sequence 3, 9, 27, 81…….
Find the next three terms of the sequence than write a rule to describe
the sequence.
3
9
x 3
27
x 3
81
x3
243
X 3
X3
The rule would be to Start with 3 and multiply by 3 repeatedly.
The common ratio is
3
1
. Why did we use the word “ratio?”
729
So, that’s it?????
Of course not! Not every sequence is arithmetic or geometric.
You can determine whether ANY sequence of terms is
arithmetic or geometric by looking for a common difference or
a common ratio.
The other sequences just consist of patterns designed to make
you crazy.
Finding the Type of Sequence
a. 3, 5, 9, 15, . . 23,
.
33, 45
+2 + 4 + 6
+8
continue this pattern
+ 10
+ 12
The sequence is neither arithmetic nor geometric. Following the
pattern above, the next three terms are 23, 33, and 45.
b. 2, -4, 8, -16, . . . 32, -64, 128
x (-2)
x (-2)
x(-2)
x (-2)
x (-2)
x (-2)
2
The common ratio is equal to 1 .
The sequence is geometric. The next three terms are 32, -64,
and 128.
Can anyone find the next term?
What if………
. . . . you know the sequence is geometric, but the common ratio
is some obscure number ?!?
1
1
1
1
4
6 , 5 , 4 , ...3 41
2
5
25
125
6 n5
2
5
13
26
n
2
5
You just need to find the ratio that was used to get from one
term to the next!
2
2 13
26 2
( ) n
( )
13 2
5 13
4
n
5
Back to your notes……
Find the next three terms. Write a rule to describe the
sequence. Find the common ratio for c. and d.
a. 24, 20, 16, 12, . . . 8, 4, 0
Start with 24 and subtract 4.
b.
1
2
2, 3 , 4 , 6
3
3
c. 4, 12, 36, 108
1
2
7 , 8 , 10
3
3
Start with 2 and add (1 1/3 ) repeatedly.
324, 972, 2916
Start with 4 and multiply by 3 repeatedly. The common multiplier is 3.
d. 4, 2, 1, 0.5
.25, .125, .0625
Start with 4 and multiply by .5 (1/2) repeatedly. The common ratio is ½ .
Identify each sequence as arithmetic, geometric, or
neither. Find the next three terms.
a. 3, 9, 27, 81, . . . 243, 729, 2187
Geometric
b. 10, 13, 18, 25 . . . 34, 45, 58
Neither…..we are adding consecutive odd numbers.
c. -12, 12, -12, 12, … -12, 12, -12
Geometric
. . . Well?
d. 50, 200, 350, 500
Arithmetic
650, 800, 950
What happened?????? 8x or y = 8x
x
y
1
2
3
4
8
16
24
32
Think of
(x, input)
(y, output)
The word “position” means the 1st term in the list, the 2nd term in the
list, and so on. The Value shows the term in the sequence.
Just as with a function rule, you can write a rule that shows how
the value (y) was created from the position number (x).
3x
3n will give us the output.
Read this as, “The first
term is 3, the 2nd one is
6, the 3rd one is 9…… “
What do you see??
This
identifies
the
function.
The tenth term?
6n
Each consecutive term is multiplied by 6.
6n is the expression that identifies the function.
n+4
Each consecutive term has been increased by 4.
n + 4 is the expression that identifies the function.
More than one operation was used for this
sequence. We can use this difference of 2 to help
us find the other operation.
Let’s experiment with this value to see if we
can find another pattern.
Do you see an
arithmetic or
geometric pattern?
(2) + ?
(2) + ?
(2) + ?
+2
+2
+2
2x + 3
After each value was doubled, 3 was added.
Let’s look for things
they have in
common.
Let’s try another one…..
This is not in your notes. I thought you might like to see this
process again.
Input
Output
1 (3) -
???
2
(3) 3 (3) 4 (3) -
1
4
7
10
Would 3x – 2 work for all terms?
Y = 3x – 2
+ 3 from one term to the next.
(2) +?
(2) + ?
(2)+?
The change is not just addition or multiplication.
Experiment…..
1x2=2
2+1=3
2x2=4 4+1=5
2x + 1 will work on all values.
2
2
2
The change is not just addition or multiplication.
Experiment…..
1x2=2
2+1=3
2x2=4 4+1=5
2x + 1 will work on all values.
What did we accomplish?
You tell me………….