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Unit3 #7
Name:______________________________________
Sequences
A sequence is an ordered list of numbers. Each number is called a _term__ of the sequence.
An _arithmetic sequence_ is a sequence in which the difference between any two consecutive terms
is the same.
each term is found
by _adding____ the
same number to the
previous term
___________
Arithmetic
sequence
8, 11, 14, 17, ...
"n + 3"
+3 +3 +3
The difference between any two consecutive terms in an arithmetic sequence is called the
_common_ __difference__.
State whether the sequence 3, 5, 7, 9, 11, … is arithmetic. If it is, state the common difference and
write the next three terms.
3, 5, 7, 9, 11, _13_, _15_, _17_, …
Notice, that 5 – 3 = 2
7–5= 2
9–7= 2
, and so on.
Write the rule as an algebraic expression: “The previous term n, increased by 2” ___n + 2____
Sometimes the arithmetic sequence looks like the term is subtracted, but it’s really adding a negative.
State whether the sequence is arithmetic. If it is, find the values of the following:
17, 12, 7, 2, –3, …
4, 1, –2, –5, …
–20, –16, –12, …
Common difference: _–5___
Common difference: _–3__
Common difference: _______
Next three terms: –8, –13, –18,
Next three terms: _–8, –11_, –14_ Next three terms: ____, ____, ____
Ninth term: __–23____
Eighth term: _–17_____
Seventh term: __________
Rule: _n + (– 5)__
Rule: _n + (– 3)___
Rule: __________
If the numbers are growing, then the common difference is __positive__.
If the numbers are getting smaller, then the common difference is __negative____.
Geometric
_____________
sequence
each term is found
by multiplying
term by the same
number
3, 6, 12, 24,..
"2n"
•2 •2 •2
The factor between any two consecutive terms in a geometric sequence is called the _common_
__factor__.
State whether the sequence –2, –4, –8, –16, –32, … is geometric. If it is, state the common ratio and
write the next three terms.
–2, –4, –8, –16, –32, _–64_, _–128_, _–256_, …
Notice, that
=2
=2
= 2
, and so on.
Write the rule as an algebraic expression: “The previous term n, multiplied by 2” _n ∙ 2 OR 2n_
Sometimes the geometric sequence looks like the term is divided, but it’s really multiplying a fraction.
State whether the sequence is geometic. If it is, find the values of the following:
81, 27, 9, 3, 1,
,…
Common ratio:
Next three terms: ,
1, –5, 25, –125, …
Common ratio: _–5__
,
–32, –8, –2, …
Common ratio:
Next three terms: 625, –3125, 15625 Next three terms: – , – , –
Ninth term:
Eighth term: 78125
Seventh term: –
Rule:
Rule: _–5n___
Rule:
If the signs of all terms are the same (all positive OR all negative), then the common ratio is __positive__.
If the sign change for every next term, then the common ratio is __negative____.
State if each sequence is arithmetic, geometric, or neither. If the sequence is arithmetic or geometric, fill in the table
and describe the relationship (write the rule) between two consecutive terms in the sequence.
Sequence
0, 3, 6, 9, 12, . . .
arithmetic,
geometric, or neither
arithmetic
Common
difference/ratio
3
Next three terms
Relationship
15 ,18, 21
n+3
48, 24, 12, 6, 3, . . .
geometric
6, 11, 16, 21, 26, . . .
arithmetic
5
31, 36, 41
neither
none
15, 21, 28
, 1, 3, 9, . . .
geometric
3
27, 81, 243
3n
30, 26, 22, 18, 14, . . .
arithmetic
–4
10, 6, 2
n–4
–3, –6, –12, –24, . . .
geometric
2
– 48, – 96, – 192
2n
–4, 4, –4, 4, –4, . . .
geometric
–1
4, –4, 4
–1 ∙ n
–5, 10, –20, 40, . . .
geometric
–2
– 80, 160, – 320
– 2n
neither
none
2, 1, 2
0, 1, 3, 6, 10, . . .
1, 2, 1, 2, 1, . . .
448, 224, 112, 56, . . .
, ,
geometric
n+5
28, 14, 7
1. Write a sequence for the perimeters of the rectangles.
12, 16, 20, 24
Is the sequence arithmetic or geometric? arithmetic
Common difference/ratio: __4___
Next three terms: _28_, 32, 36
Rule: n + 4
2. Write a sequence for the areas of the rectangles: 8, 15, 24, 35
Is the sequence arithmetic or geometric? neither
Common difference/ratio: none
Next three terms: 48, 63, 80
Rule: add 2 more than we added the
previous time
3. Write an arithmetic sequence in which the common difference is –5. ____________________________________
HOMEWORK 3.7
Sequence
arithmetic,
geometric, or neither
Common
difference/ratio
Next three terms
Relationship
35, 28, 21, 14…
1, 3, 9, 27…
2, –4, 8, –16…
121, 1221, 12221 …
, 1, –2, 4, . . .
–8, –6, –4, –2…
2, 6, 24, 120…
–2, –6, –18, –54 …
8) What is the 7th term of the arithmetic sequence:
29 ,
21 ,
13 ,
5,
_______
9) Find the missing terms in the sequence 26, 33, 40, ___, 54, 61, ___, …
13) Create three sequences. One sequence must be arithmetic, one sequence must be geometric, and one
sequence must be neither. Then describe the relationship between two consecutive terms in each of your
arithmetic and geometric sequences.
Arithmetic: ________________________________________ ______________
Geometric: ________________________________________ ______________
Neither:
________________________________________
Look for a pattern. Then use the pattern to solve each problem.
There were 256 people at a fundraiser. When the event was over, half of the people who remained left every 5 minutes.
How long after the event ended did the last person leave?
A knitting shop is having a huge yarn sale. One skein sells for $1.00, 2 skeins sell for $1.50, and 3 skeins sell for $2.00. If
this pattern continues, how many skeins of yarn can you buy for $5.00?
The football cheerleaders will arrange themselves in rows to form a pattern on the football field at halftime. In the first
five rows there are 12, 10, 11, 9, and 10 girls in each row. They will form a total of twelve rows. If the pattern continues,
how many girls will be in the back row?
Find the perimeters of the next two figures in the pattern. The length of
each side of each small square is 3 feet.
A hot tub holds 630 gallons of water when it is full. A hose fills the tub at a
rate of 6 gallons every five minutes. How long will it take to fill the hot tub?
Jordan saved $1 the first
week, $2 the second week, $4 the third
week, and $8 the fourth week. If this
pattern continues, how much will she
save the eighth week?