1.  DONE BY,
‘
TN MBHAMALI
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2. SEQUENCES
AND SERIES
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8. An introduction…………
1, 4, 7, 10, 13
9, 1,
7,
3,
6
2, 4, 8, 16, 32
2 7 .2
3
62
9,
12
15
6.2, 6.6, 7, 7.4
,
35
20 / 3
3, 1,
1/ 3
1, 1 / 4, 1 / 16, 1 / 64 8 5 / 6 4
9
, 2.5 , 6.25
Arithmetic Sequences
Geometric Sequences
ADD
To get next term
9 .7 5
MULTIPLY
To get next term
Arithmetic Series
Sum of Terms
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Geometric Series
Sum of Terms
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9. Find the next four terms of –9, -2, 5, …
Arithmetic Sequence
2
9
5
2
7
7 is referred to as the common difference (d)
Common Difference (d) – what we ADD to get next term
Next four terms……12, 19, 26, 33
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10. Find the next four terms of 0, 7, 14, …
Arithmetic Sequence, d = 7
21, 28, 35, 42
Find the next four terms of x, 2x, 3x, …
Arithmetic Sequence, d = x
4x, 5x, 6x, 7x
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -32k
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11. Vocabulary of Sequences (Universal)
a1
F irst te rm
an
n th te rm
n
Sn
d
num ber of term s
sum of n term s
com m on difference
nth term of arithm etic sequence
an
sum of n term s of arithm etic sequen ce
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a1
n
Sn
1 d
n
2
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a1
an
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12. Given an arithmetic sequence with a 15
x
a1
an
n th te rm
15
n
NA
Sn
3, find a 1.
F irst te rm
38
38 and d
-3
d
num ber of term s
sum of n term s
com m on difference
an
a1
n
38
x
15
1 d
1
3
X = 80
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13. Find S 63 of
19,
13, 7,...
-19
F irst te rm
??
an
n th te rm
63
353
a1
n
x
an
??
??
a1
19
n
Sn
6
num ber of term s
d
sum of n term s
com m on difference
1 d
63
Sn
1
6
S 63
353
S 63
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n
2
a1
63
an
19
353
2
1 052 1
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14. Try this one:
Find a 16 if a 1
1.5 and d
0.5
1.5
a1
F irst te rm
x
an
n th te rm
16
n
NA
Sn
0.5
d
num ber of term s
sum of n term s
com m on difference
an
a 16
1.5
a 16
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a1
9
TN MBHAMALI
n
16
1 d
1 0. 5
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15. F ind n if a n
633, a 1
9, and d
24
9
a1
F irst te rm
633
an
n th te rm
x
n
NA
Sn
24
d
num ber of term s
sum of n term s
com m on difference
an
a1
n
1 d
633
9
x
1 24
633
9
24x
24
X = 27
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16. Find d if a 1
6 and a 29
20
-6
a1
F irst te rm
20
an
n th te rm
29
n
NA
Sn
x
d
an
num ber of term s
sum of n term s
com m on difference
a1
n
20
6
29
26
28 x
x
1 d
1 x
13
14
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17. Find two arithmetic means between –4 and 5
-4, ____, ____, 5
-4
a1
F irst te rm
5
an
n th te rm
4
n
NA
x
num ber of term s
Sn
sum of n term s
d
com m on difference
an
a1
n
1 d
5
4
4
1 x
x
3
The two arithmetic means are –1 and 2, since –4, -1, 2, 5
forms an arithmetic sequence
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18. Find three arithmetic means between 1 and 4
1, ____, ____, ____, 4
1
a1
F irst te rm
4
an
n th te rm
5
n
NA
x
num ber of term s
Sn
sum of n term s
d
com m on difference
an
a1
4
1
x
n
5
1 d
1 x
3
4
The three arithmetic means are 7/4, 10/4, and 13/4
since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence
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19. Find n for the series in which a 1
5
a1
F irst te rm
y
an
n th te rm
x
n
5, d
440
Sn
3
d
an
y
440
3, S n
x
5
440
x 7
x
1 3
com m on difference
1 d
3x
2
sum of n term s
n
5
2
num ber of term s
a1
440
880
0
x 7
3x
2
3x
7x
880
Graph on positive window
5
x
1 3
X = 16
Sn
440
n
2
x
a1
5
an
y
2
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20. 05/03/2014
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21. An infinite sequence is a function whose domain
is the set of positive integers.
a1, a2, a3, a4, . . . , an, . . .
terms
The first three terms of the sequence an = 2n2 are
a1 = 2(1)2 = 2
a2 = 2(2)2 = 8
finite sequence
a3 = 2(3)2 = 18.
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22. A sequence is geometric if the ratios of consecutive
terms are the same.
2, 8, 32, 128, 512, . . .
8
2
32
8
4
512
128
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4
128
32
geometric sequence
4
4
The common ratio, r, is 4.
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23. The nth term of a geometric sequence has the form
an = a1rn - 1
where r is the common ratio of consecutive terms of
the sequence.
r
a1 = 15
75
15
5
15, 75, 375, 1875, . . .
a2 =
15(5)
a3 =
a4 =
15(52) 15(53)
The nth term is 15(5n-1).
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24. Example: Find the 9th term of the geometric sequence
7, 21, 63, . . .
a1 = 7
r
21
7
3
an = a1rn – 1 = 7(3)n – 1
a9 = 7(3)9 – 1 = 7(3)8
= 7(6561) = 45,927
The 9th term is 45,927.
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25. The sum of the first n terms of a sequence is
represented by summation notation.
upper limit of summation
n
ai
a1
a2
a3
a4

an
i 1
lower limit of summation
index of summation
5
4
n 1
n
4
4
1
4
16
2
4
64
3
4
4
256
4
5
1024
1364
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26. The sum of a finite geometric sequence is given by
n
Sn
a1r
n
a1 1 r .
1 r
i 1
i 1
5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?
n=8
r
a1 = 5
n
Sn
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a1 1 r
1 r
8
5 1 2
1 2
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5 1
256
1 2
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10
5
2
5
255
1
1275
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27. The sum of the terms of an infinite geometric
sequence is called a geometric series.
If |r| < 1, then the infinite geometric series
a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .
has the sum S
a1r
i 0
If r
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a1
i
1
r
.
1 , then the series does not have a su m.
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28. Example: Find the sum of 3 1 1
3
a1
1

1
3
r
S
1
9
3
r
1
3
1
3
1
1
3
3
4
3
3 3
4
9
4
The sum of the series is 9 .
4
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29. 
Geometric Sequences and Series
A geometric sequence is a sequence in which each term after the first is
obtained by multiplying the preceding term by a constant nonzero real
number.
1, 2, 4, 8, 16 … is an example of a geometric sequence with first term
1 and each subsequent term is 2 times the term preceding it.
The multiplier from each term to the next is called the common ratio
and is usually denoted by r.
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30. 
Finding the Common Ratio
In a geometric sequence, the common ratio can be found by dividing any
term by the term preceding it.
The geometric sequence 2, 8, 32, 128, …
has common ratio r = 4 since
8
128
2
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32
8
32
TN MBHAMALI
...
4
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31. 
Geometric Sequences and Series
nth Term of a Geometric Sequence
In the geometric sequence with first term a1 and
common ratio r, the nth term an, is
an
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a1 r
n 1
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32. 
Using the Formula for the nth Term
Example Find a5 and an for the geometric
sequence 4, –12, 36, –108 , …
Solution Here a1= 4 and r = 36/ –12 = – 3. Using
n 1
n=5 in the formula a n a1 r
a5
4 ( 3)
5 1
4 ( 3)
4
324
In general
an
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a1 r
n 1
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4 ( 3)
n 1
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33. 
Modeling a Population of Fruit Flies
Example A population of fruit flies grows in such a
way that each generation is 1.5 times the previous
generation. There were 100 insects in the first
generation. How many are in the fourth generation.
Solution The populations form a geometric sequence
with a1= 100 and r = 1.5 . Using n=4 in the formula
for an gives
a4
a1 r
3
100(1.5)
3
337.5
or about 338 insects in the fourth generation.
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34. 
Geometric Series
A geometric series is the sum of the terms of a
geometric sequence .
In the fruit fly population model with a1 = 100 and r
= 1.5, the total population after four generations is a
geometric series:
a1
a2
100
a3
a4
100(1.5)
100(1.5)
2
100(1.5)
3
813
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35. 
Geometric Sequences and Series
Sum of the First n Terms of an Geometric
Sequence
If a geometric sequence has first term a1 and common
ratio r, then the sum of the first n terms is given by
Sn
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a1 (1
1
n
r )
r
TN MBHAMALI
where
r
1
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36. 
Finding the Sum of the First n Terms
6
Example Find
2 3
i
i 1
Solution This is the sum of the first six terms of a
1
6 and r = 3.
geometric series with a1 2 3
From the formula for Sn ,
S6
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6 (1
1
6
3 )
6 (1
3
TN MBHAMALI
729)
2
6 ( 7 2 8)
2
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.
36

37. Vocabulary of Sequences (Universal)
a1
F irst te rm
an
n th te rm
n
num ber of term s
Sn
r
sum of n term s
com m on ratio
nth term of geom etric sequence
an
sum of n term s of geom etric sequ ence
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a 1r
n 1
Sn
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a1 r
r
n
1
1
37

38. Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic
3
9/2
2
3
1 .5
g e o m e tric
3
r
2
9 9
2, 3, ,
2 2
3 9
,
2 2
3
2
3 9
,
2 2
3
3
3
2
2
2
9 27 81 243
2, 3, ,
,
,
2 4 8 16
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39. If a 1
1
2
,r
2
3
, fin d a 9 .
a1
an
n th te rm
x
num ber of term s
r
NA
com m on ratio
Sn
9
sum of n term s
n
2/3
an
a 1r
n 1
1
2
9 1
3
2
x
TN MBHAMALI
2
2
x
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1/2
F irst te rm
8
3
2
8
7
3
8
128
6561
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40. Find two geometric means between –2 and 54
-2, ____, ____, 54
a1
an
n
Sn
r
-2
F irst te rm
54
n th te rm
num ber of term s
sum of n term s
com m on ratio
4
an
a 1r
54
NA
n 1
2
x
27
3
x
x
4 1
3
x
The two geometric means are 6 and -18, since –2, 6, -18, 54
forms an geometric sequence
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41. F in d a 2
a 4 if a 1
3 and r
2
3
-3, ____, ____, ____
2
S in c e r
...
3
3,
2,
4
8
,
3
a2
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a4
2
TN MBHAMALI
9
8
9
10
9
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42. F in d a 9 o f
2 , 2, 2 2 ,...
a1
F irst te rm
an
n th te rm
x
num ber of term s
9
n
Sn
r
2
NA
sum of n term s
com m on ratio
r
an
a 1r
2
2
n 1
2 2
2
2
9 1
x
2
2
8
x
x
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2
16 2
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2
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43. If a 5
32 2 and r
2 , find a 2
____, ____ , ____, ____ ,32 2
a1
an
x
F irst te rm
n th te rm
n
Sn
r
32 2
num ber of term s
NA
sum of n term s
2
com m on ratio
an
a 1r
5
n 1
5 1
32 2
x
2
4
32 2
x
32 2
4x
8 2
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2
x
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44. *** Insert one geometric mean between ¼ and 4***
*** denotes trick question
1
, ____, 4
4
a1
F irst te rm
1/4
an
n th te rm
4
num ber of term s
3
n
sum of n term s
NA
com m on ratio
x
Sn
r
an
4
1
4
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r
3 1
4
1
r
2
a 1r
16
r
2
1
n 1
4
4
r
1
4
TN MBHAMALI
, 1, 4
,
1, 4
4
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45. F in d S 7 o f
1
1
1
2
4
8
...
a1
F irst te rm
1/2
an
n th te rm
NA
n
r
Sn
a1 r
r
x
1
r
com m on ratio
1
4
1
8
1
4
1
2
1
1
2
7
2
1
2
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sum of n term s
1
1
x
7
2
Sn
n
num ber of term s
1
1
2
1
7
2
1
63
1
1
64
2
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46. Reference list
http://www.slideshare.net/bercando/sequence-and-series-11636098?qid=9878037e-6028-46c4-9246e3fb4a648965&v=qf1&b=&from_search=11
http://www.slideshare.net/mstfdemirdag/sequences-and-series-11032997?qid=5fcd86dc-7287-4e52972a-c8392094a20d&v=qf1&b=&from_search=6
https://www.google.co.za/#q=what+is+a+sequence
http://www.slideshare.net/jfuller2012/sequences-and-series-6125259?qid=9878037e-6028-46c49246-e3fb4a648965&v=qf1&b=&from_search=12
http://www.slideshare.net/mcatcyonline/sequence-and-series-1902957?qid=9878037e-6028-46c49246-e3fb4a648965&v=default&b=&from_search=15
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