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Arithmetic Sequences 2013 14
Accountancy (Hjkk10)
Kalayaan College
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####### Grade Level/Course: Algebra
####### Lesson/Unit Plan Name: Arithmetic Sequences
####### Rationale/Lesson Abstract: Students will be introduced to sequences and learn the
####### characteristic that make sequences arithmetic. In addition, students will write the recursive
####### and explicit formulas by analyzing patterns. Lastly, students will make connections between
####### arithmetic sequences and functions.
####### Timeframe: Two class periods
####### Common Core Standard(s): F-‐BF Write arithmetic and geometric sequences both recursively
####### and with an explicit formula, use them to model situations, and translate between the two
####### forms.
####### Note: the Warm-‐Up is on page 10.
Instructional Resources/Materials: Warm-‐Up, Mix and Match Activity Cards
####### Lesson:
**Think -‐Pair -‐Share: ** Can you find a pattern and use it to guess the next term?
####### A) ,7 10 , 13 , 16 ...,
####### B) 14 − ...,4,2,8,
####### C) ,9,4,1 16 ...,
TPS Goal: Students notice that the terms in the first sequence are increasing by 3, the terms in the sec ond sequence are decreasing by 6, and the terms in the third are not increasing by a constant.
A) 19 or a 5 = 19 B) − 10 or a 5 =− 10 C) 25 or a 5 = 25
Example 1: Find the missing terms in each sequence.
####### a) ,,, ...,, aaaa 125321 ,,... b) 321 ,,, ...,, aaaa n ,,...
####### 321 ,,, ..., 125 − 1 ,, aaaaaa 125125 + 1 ,... ,,, ..., − aaaaaa nnn + 11321 ,,, ...
321 ,,, ..., ,, aaaaaa 126125124 ,...
Note: If d > 0, then the terms of the sequence are increasing, and if d < 0, then the terms are decreasing.
**Think -‐Pair -‐Share: ** Determine if each sequence is arithmetic. If yes, identify the common difference.
####### A) ,7 10 , 13 , 16 ...,
####### B) 14 − ...,4,2,8,
####### C) ,9,4,1 16 ...,
####### Answers: A) Yes, d = 3 B) Yes, d =− 6 C) No
A sequence is a list or an ordered arrangement of numbers, figures or objects. The members, which are also elements, are called the term s of the sequence. A general sequence can be written as
####### aaaaaa 654321 ,,,,,, ...
where a 1 is the first term, a 2 is the second term, and so on. The nth term is denoted as an .
An arithmetic sequence is a list of numbers in which the difference between two consecutive terms is constant. The common difference is called d .
Example
3: Write the recursive formula for the sequence ,7 10 , 13 , 16 ,...
The formula depends on the common difference which we already identified from the previous
####### example. Substitute d = 3 into nn − 1 += daa and identify a 1 . The recursive formula for this sequence
is aa nn − 1 += 3 where a 1 = 7 .
TRY: Find the next term and write the recursive formula for each sequence.
Partner
A: 75 , 87 , 99 , ... 111 , nn − 12 , aaaa 114 =+== 75
Partner
B: 27 , 21 , 15 ,3,9, ... =− = nn − − ,6,3 aaaa 116 = 27
**Discuss: ** Why is it necessary to identify a 1 in the recursive formula?
If a 1 is not identified, then the formula represents any sequence who has the same common difference.
For
example, the sequences ,7 10 , 13 , 16 ,... and ,8,5,2 11 ..., have the same common difference but
the recursive formula for the second sequence is aa nn − 1 += 3 where a 1 = 2.
Example
4: Find the 50th term in the sequence ,7 10 , 13 , 16 ,...
Discuss: We can use the recursive formula repeatedly to obtain desired terms. Or i s there another way?
Decompose each term to find a pattern: Separate each term:
( ) ( ),337,237,37, ...
####### ,3337,337,37, ...
####### ,37,7 10 ,3 13 ,3...
####### ,7 10 , 13 , 16 ,...
####### +++
####### ++++++
####### +++
( )
( )
( )
37 ( ) 49
####### ...
####### 337
####### 237
####### 137
####### 7
50
4
3
2
1
####### +=
####### +=
####### +=
####### +=
####### =
####### a
####### a
####### a
####### a
####### a
####### ∴ a 50 = 154
Any term in a sequence can be found with an explicit formula, which does not depend on the previous
####### term. An explicit formula is a formula that expresses any term an in terms of n , its position in the
sequence.
Recall: ( ) xf mx += b is
a formula, more specifically a function, written in terms of x .
Look for a pattern:
How many groups of 3 are added to 7 in the 50 th term?
Answer:
Give other examples, if necessary, to build from prior knowledge.
Example 5: Write the explicit formula for the sequence. Then find a 100 .
,7 10 , 13 , 16 ,...
Decompose each term to find a pattern: Separate each term:
( ) ( ),337,237,37, ...
####### ,3337,337,37, ...
####### ,37,7 10 ,3 13 ,3...
####### ,7 10 , 13 , 16 ,...
####### +++
####### ++++++
####### +++
( )
( )
( )
( )
( ) 137
####### 37 49
####### ...
####### 337
####### 237
####### 137
####### 7
50
4
3
2
1
####### += −
####### +=
####### +=
####### +=
####### +=
####### =
####### na
####### a
####### a
####### a
####### a
####### a
n
Explicit Formula n += ( ) na − 137 or
n na += 43
Find a 100 ( )
( )
( )
304
7 297
37 99
37 100 1
137 100
=
+=
+=
+= −
+= − a
n na
or
( )
304
300 4
3100 4
43 100
=
+=
+=
+= a
n na
∴ a 100 = 304
TRY: Write
the explicit formula for each sequence. Then use it to find a 30 .
Partner
A: 75 , 87 , 99 , ...
( )( )
423
6312or
75 112
30 =
+=
+= −
a
na
na n
n
Partner
B: 27 , 21 , 15 ,3,9, ...
( )( )
147
or 6 33
27 61
30 =−
=− +
+= −−
a
na
na n
n
The
explicit formula (or rule) for an arithmetic sequence is n 1 += ( )− 1 dnaa .
**Mix and Match Activity: ** Each student receives a card. When prompted, students are to seek the person who has a card that matches their card. Students should be asking appropriate questions using the mathematical language presented in the lesson to find their match. When students find their match, you may have them do an extension activity. Extensions may include, writing about their problem, making a poster/foldable, gallery walk, or reshuffling the cards and repeating the process.
Copy and cut the tables on the next two pages. The corresponding cards are matched for you to use an answer key.
Exit Ticket: Determine if each statement is true or false.
####### 1) The sequence ,8,5,3,2,1,1 11 ..., is arithmetic.
2) The sequence ,... 2
, 1 2
, 1 2
, 3 2
5 − is arithmetic.
3) The recursive formula nn − aaa 11 =+= 1,2 represents the sequence
####### ,5,3,1 10 ...,7.
####### 4) The explicit formula n =− 2 na + 15 represents the sequence 15 , 13 , 11 ...,9,
5) The recursive formula nn − ,8 aaa 11 =+= 12 and the explicit formula n na += 48 represent the same sequence.
ANSWERS: F, T, T, F, T
! True! False !!
! True! False !!
! True! False !!
! True! False !!
! True! False !!
Is the sequence arithmetic?
####### ,9,7,5 11 ,−−−−− 15 ,...
If yes, identify the common difference.
No, the difference between consecutive terms is not constant.
Write the recursive formula for the sequence:
####### ,...5,2,5,1,5.
nn − aaa 11 =+= 5,5.
Write the explicit formula for the sequence:
####### ,..,7,5,3,
n += ( ) na − 211
Does the recursive formula nn − aaa 11 =+= 0,
####### represent the sequence ...,125,25,5,0 ?
No, the difference between consecutive terms is not 5.
Does
the explicit formula n += ( ) na − 413
####### represent the sequence ,7,3 11 , 15 ..., ?
Yes, the first term is 3 and the common difference is 4.
Find the sequence who has the recursive formula
nn − aaa 11 =+= −9,4 and the explicit formula n = 4 na − 13 .
####### −−− ,..,3,1,5,
Write the explicit formula for an arithmetic sequence where d = 2 and a 4 = 30 .
n 24 += ( ) na − 21
Identify the first five terms of an arithmetic sequence whose common difference is 10.
####### 33, 43, 53, 63, 73,...
Warm-Up
CCSS :
F-‐ IF 2 CCSS: F-‐IF 4
Current:
Current continued:
_x _
_y _
Given
the linear function ( )= xxf − 52 ,
indicate
whether each statement is true
or
false.
A) f ( )− 3 =− 11
B) f ( )= 50
C) 4
2
1
⎟⎟=
⎠
⎞ ⎜⎜⎝
⎛
f
D)
( )=− + ( ) xxf − 123
Identify
the correct outputs for the false
statements.
Graph
the linear function ( ) xxg += 1 and
complete
the statements below.
a) ( ) xg < 0 when
x ______ .
b) ( ) xg = 0 when
x ______.
c) ( ) xg > 0 when
x ______.
Write
your own true statement about
( ) xg .
! True! False !!
! True! False !!
! True! False !!
! True! False !!
Suppose
a movie theater has 42 rows of
seats
and there are 29 seats in the first
row.
Each row after the first row has
two
more seats than the row before it.
Fill
in the table below to find the number
of
seats in each row.
Row
Number Number of Seats
1
2
3
4
5
10
30
42
Discuss
with a partner how you
completed
the table. Is there another
method?
Warm-Up
Solutions
CCSS: F-‐ IF
2 CCSS: F-‐IF 4
Current:
Other:
_x _
_y _
Given
the linear function ( )= xxf − 52 ,
indicate
whether each statement is true
or
false.
A) f ( )− 3 =− 11
B) f ( )= 50
C) 4
2
1
⎟⎟=
⎠
⎞ ⎜
⎜ ⎝
⎛
f
D)
( )=− + ( ) xxf − 123
Corrections:
f ( )=− 50 and 4
2
1
⎟⎟=−
⎠
⎞ ⎜⎜⎝
⎛
f
Graph
the linear function ( ) xxg += 1 and
complete
the statements below.
a) ( ) xg < 0 when
x <− 1 .
b) ( ) xg = 0 when
x =− 1 .
c) ( ) xg > 0 when
x >− 1 .
Write
your own true statement about
( ) xg .
Sample
Answer: g ( )= 10
! True! False !!
! True! False !!
! True! False !!
! True! False !!
Suppose
a movie theater has 42 rows of
seats
and there are 29 seats in the first
row.
Each row after the first row has
two
more seats than the row before it.
Fill
in the table below to find the number
of
seats in each row.
Row
Number Number of Seats
1
29
2
31
3
33
4
35
5
37
10
47
30
87
42
111
Discuss
with a partner how you
completed
the table. Is there another
method?
Sample
Answer: Instead of finding the
number
of seats in row 6-‐9 to get row
10,
understand that you will be adding
5
groups of 2 seats (totaling 10 seats)
from
row 5 to 10. From row 10 to 30,
you
will be adding 20 groups of 2 seats
(40
seats). And from row 30 to 42,
you
will be adding 12 groups of 2 seats
(24
seats).
Arithmetic Sequences 2013 14
Course: Accountancy (Hjkk10)
University: Kalayaan College
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