arithmetic sequences
DESCRIPTION
Arithmetic Sequences. U SING AND W RITING S EQUENCES. The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1. - PowerPoint PPT PresentationTRANSCRIPT
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Arithmetic Sequences
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USING AND WRITING SEQUENCES
The numbers in sequences are called terms.
You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.
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The domain gives the relative position of each term.
1 2 3 4 5 DOMAIN:
3 6 9 12 15RANGE:The range gives the terms of the sequence.
This is a finite sequence having the rulean = 3n,
where an represents the nth term of the sequence.
USING AND WRITING SEQUENCES
n
an
3
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Writing Terms of Sequences
Write the first six terms of the sequence an = 2n + 3.
SOLUTION
a 1 = 2(1) + 3 = 5 1st term
2nd term
3rd term
4th term
6th term
a 2 = 2(2) + 3 = 7
a 3 = 2(3) + 3 = 9
a 4 = 2(4) + 3 = 11
a 5 = 2(5) + 3 = 13
a 6 = 2(6) + 3 = 15
5th term
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Writing Terms of Sequences
Write the first six terms of the sequence f (n) = (–2) n – 1 .
SOLUTION
f (1) = (–2) 1 – 1 = 1 1st term
2nd term
3rd term
4th term
6th term
f (2) = (–2) 2 – 1 = –2
f (3) = (–2) 3 – 1 = 4
f (4) = (–2) 4 – 1 = – 8
f (5) = (–2) 5 – 1 = 16
f (6) = (–2) 6 – 1 = – 32
5th term
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An introduction…………
1, 4, 7,10,139,1, 7, 156.2, 6.6, 7, 7.4, 3, 6
ARITHMETIC
ADD(by the same #)
To get the next term
2, 4, 8,16, 329, 3,1, 1/ 31,1/ 4,1/16,1/ 64
, 2.5 , 6.25
GEOMETRIC
MULTIPLY(by the same #)
To get the next term
d = 3 d = -8 d = .4 d = 3
r =2
r = 41
5.2r =
6
r = 31
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Vocabulary of Sequences (Universal)
1a First term
na nth term
n number of termsd common difference
r common ratio
Finite VS. Infinite 7
an-1 previous term an+1 next term
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Arithmetic Sequence: sequence whose consecutive terms have a common difference.
Example: 3, 5, 7, 9, 11, 13, ...
The terms have a common difference of 2. (known as d)
To find the common difference you use an+1 – an Example: Is the sequence arithmetic? If so, find d.
–45, –30, –15, 0, 15, 30 d = 15
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Find the next 4 terms of –9, -2, 5, …
2 9 5 2 7 7 is referred to as d
Next four terms…… 12, 19, 26, 33
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Arithmetic Sequence, d = 7 21, 28, 35, 42
Arithmetic Sequence, d = x 4x, 5x, 6x, 7x
Find the next four terms of 0, 7, 14, …
Find the next four terms of x, 2x, 3x, …
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -31k10
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The nth term of an arithmetic sequence is given by:
1 ( 1)na a n d The nth term in the sequence
First term
The common difference
The term #
)6(346664)6(24664)6(1464)6(044
4
3
2
1
aaaa
4, 10, 16, 22
585446)110(410 aFind the 10th term:
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Find the 14th term of the sequence: 4, 7, 10, 13,
……1 ( 1)na a n d
14a 4 (13)3 43
3)114(4
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1 ( 1)na a n d
In the arithmetic sequence 4,7,10,13,…, which term has a value of 301?
301 4 ( 1)3n 301 4 3 3n 301 1 3n 300 3n 100 n
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Given an arithmetic sequence with 15 1a 38 and d 3, find a .
n 1a a n 1 d
38 x 1 15 3
X = 8014
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1 29Find d if a 6 and a 20
120 6 29 x
26 28x13x14
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Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence?
an = a1 + (n – 1)d d = 4, a5 = 15, n = 5, a1=?
15 = a1 + (5 – 1)4 15 = a1 +16 a1 = –1
a10 = –1 + (10 – 1)4= -1 + 36
a10 = 35 16
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Explicit Formula – used to find the nth term of the arithmetic sequence in which the common difference and 1st term are known.
Ex: 4, 6, 8, 10…
Use a1 and d in sequence formula: an = 4 + (n – 1)2 an = 2n + 2
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Explicit vs. Recursive Formulas
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Find the explicit formula for the following arithmetic sequence:3, 8, 13, 18…
an = a1 + (n – 1)d a1 = 3 d = 5 n = ?
an = 3 + (n – 1)5 an = 3 + 5n – 5
an = -2 + 5n OR an = 5n – 2 18
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Recursive Formula – (includes a1) used to find the next term of the sequence by adding the common difference to the previous term.
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Explicit vs. Recursive Formulas
an = an-1 + 2 a1 = 4Ex: 4, 6, 8, 10…
an = an-1 + da1 = ___
an+1 = an + d a1 = ___
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Find the recursive formula for the following arithmetic sequence:3, 8, 13, 18…
an = an-1 + d a1 = 3 d = 5
an = an-1 + 5 a1 = 3
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Using Recursive & Explicit Formulas
an = an-1 + 6 a1 = 4
1. Create the 1st 5 terms:4, 10, 16, 22, 28
2. Find the explicit formula:
an = a1 + (n – 1)dan = 4 + (n – 1)6an = 4 + 6n – 6 an = 6n – 2
a2 = 4 + 6 = 10 a3 = 10 + 6 = 16 a4 = 16 + 6 = 22 a5 = 22 + 6 = 28
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Using Recursive & Explicit Formulas
an = an-1 – 2 a1 = 5
1. Create the 1st 5 terms:5, 3, 1, –1, –3
2. Find the recursive formula:
an = 7 – 2n
a2 = 7 – 2(2) = 3
a5 = 7 – 2(5) = –3 a4 = 7 – 2(4) = –1 a3 = 7 – 2(3) = 1
a1 = 7 – 2(1) = 5
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An arithmetic mean of two numbers, a and b, is simply their average. Use the formula and information given to find the common difference to create the sequence. Insert 3 arithmetic
means between 8 & 16.
16 8 (5 1)d 2d
Let 8 be the 1st termLet 16 be the 5th termLet 5 be Nd is missing
1 ( 1)na a n d
8, , , ,1610 12 14
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Find two arithmetic means between –4 and 5 -4, ____, ____, 5
n 1a a n 1 d 15 4 4 x
x 3The two arithmetic means are –1 and 2,
since –4, -1, 2, 5 forms an arithmetic sequence24
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Find 3 arithmetic means between 1 & 41, ____, ____, ____, 4
n 1a a n 1 d 4 1 x15
3x4
The 3 arithmetic means are
since 1, ,4 forms a sequence4
13,4
10,47
413,
410,
47
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Geometric Sequences
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Vocabulary of Sequences (Universal)
1a First term
na nth term
n number of termsd common difference
r common ratio
an-1 previous term an+1 next term
27Finite VS. Infinite
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Find the next 3 terms of 2, 3, 9/2, __, __, __
3 – 2 vs. 9/2 – 3… not arithmetic
3 9 / 2 31.5 geometric r2 3 2
• Use to determine common ration
n
aa 1
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23
23
232
4th term:
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n 1n 1a a r
The nth term of a geometric sequence is given
by:
23
23
23
232
5th term:
23
23
23
23
232
6th term:
1st term: 2
3232 : term2nd
29
23
232 : term3rd
How is the formula derived?
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1 91 2If a , r , find a .2 3
n 1n 1a a r
9 11 2x2 3
8
8
2x2 3
7
8
23
1286561
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2 4 12Find a a if a 3 and r3
2Since r ...3
2 48 10a a 2
9 9
31
-3, ____, ____, ____2 34
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9Find a of 2, 2, 2 2,...n 1
n 1a a r 9 1x 2 2
8x 2 2
x 16 232
r = 2a1= 2
n = 9
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5 2If a 32 2 and r 2, find a ____, , ____,________ ,32 2
n 1n 1a a r
5 132 2 x 2
432 2 x 2
32 2 x4
8 2 x
21 2282 a
33
1648
2281
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Ex: 4, 12, 36, 108…
Use a1 and r in sequence formula:
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Explicit vs. Recursive FormulasExplicit Formula – used to find the nth term of the geometric sequence in which the common ratio and 1st term are known.
Ex: an = a1*rn-1 an = 4 * 3n-1
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Find the explicit formula for the following geometric sequence:3, 6, 12, 24…
an = a1*rn-1 a1 = 3 r =2
an = 3 *2n-1
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Explicit vs. Recursive Formulas
an = an-1 (–4) a1 = –1
Ex: –1, 4, –16, 64 …
an = an-1 (r)a1 = ___
an+1 = r(an) a1 = ___
Recursive Formula (includes a1) – used to find the next term of the sequence by multiplying the common ratio to the previous term.
a1 (r) = a2 a2 (r) = a3a3 (r) = a4
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Find the recursive formula for the following geometric sequence:3, 6, 12, 24…an = an-1 * r a1 = 3 r = 2
an = an-1 * 2 a1 = 3
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Using Recursive & Explicit Formulas
an = an-1 (3) a1 = –1
1. Create the 1st 5 terms:–1, –3, –9, –27, – 81
2. Find the explicit formula:
an = a1 (r)n-1
an = –1(3)n-1
a2 = –1(3) = –3a3 = –3(3) = –9 a4 = –9(3) = –27 a5 = –27(3) = –81
an = –3n-1
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Using Recursive & Explicit Formulas
an = 4an-1
a1 = 2
1. Create the 1st 5 terms:2, 8, 32, 128, 5122. Find the recursive formula:
an = 2(4)n – 1
a2 = 2(4)2-1 = 8
a5 = 2(4)5-1 = 512a4 = 2(4)4-1 = 128a3 = 2(4)3-1 = 32
a1 = 2(4)1-1 = 2
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Ex: Find two geometric means between –2 and 54
-2, ____, ____, 54
n 1n 1a a r 1454 2 x
327 x 3 x
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A geometric mean(s) of numbers are the terms between any 2 nonsuccessive terms of a geometric sequence. Use the terms given to find the common ratio and find the missing terms called the geometric means.
The 2 geometric means are 6 and -18
6 –18
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*** Insert one geometric mean between ¼ and 4****** denotes trick question
1,____,44
n 1n 1a a r
3 1144
r 2r144
216 r 4 r 1,1, 44
1, 1, 44
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Series42
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Vocabulary of Sequences (Universal)
1a First term
na nth term
n number of termsd common difference
r common ratio
an-1 previous term an+1 next term
43Finite VS. Infinite
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USING SERIES
. . .
FINITE SEQUENCE
FINITE SERIES3, 6, 9, 12, 15
3 + 6 + 9 + 12 + 15
INFINITE SEQUENCE
INFINITE SERIES
3, 6, 9, 12, 15, . . .
3 + 6 + 9 + 12 + 15 + . . .
When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite.
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You can use summation notation to write a series. For example, for the finite series shown above, you can write
3 + 6 + 9 + 12 + 15 = ∑ 3i5
i = 1
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B
nn A
a
UPPER BOUNDTERM NUMBER
LOWER BOUNDTERM NUMBER
SIGMA(SUM OF TERMS)
NTH TERMSEQUENCE
(EXPLICIT FORMULA)
45
# of Terms: B – A + 1
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j
4
1
j 2
21 2 2 3 2 24
18
7
4a
2a
42 2 5 2 6 72
4446
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An arithmetic series is a series associated
with an arithmetic sequence.
It can be infinite or finite.
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1, 4, 7, 10, 13, ….Infinite Arithmetic
(constantly getting larger or smaller)
3, 7, 11, …, 51 Finite Arithmetic n 1 nnS a a2
1, 2, 4, …, 64
1, 2, 4, 8, …
1 1 13,1, , , ...3 9 27
No Sum
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Find the sum of the 1st 100 natural numbers.
1 + 2 + 3 + 4 + … + 100
12n nnS a a
1 1a 100na 100n 100
100 (1 100)2
S
505049
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Find the sum of the 1st
14 terms of the series: 2 + 5 + 8 + 11 + 14 + 17 +…
a14 = 2 + (14 - 1)(3) = 41
301 S14 = 4122
14
50
1414 22
14 aS
1 2a 14n nn aanS 12
To find a14
, you need 3d da )114(214
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13
1
(4 5)n
n
Find the sum of the series
9 13 17 ....
1 9a 4d 13n
13(66)
2 429
12n nnS a a
1313 92
13 aS
Need 13th term:
4(13) + 5 = 57
51
5792
1313 S
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n = 4 a1 = 3 a4 = 6
12n nnS a a
18)9(26324
4 S
j
4
1
j 2
7
4a
2a
Finding the Sum from Summation Notation
n = (7 – 4) + 1 a4 = 8 a7 = 14
44)22(214824
4 S52
3, 4, 5, 6
8, 10, 12, 14
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527
2
x
3
7
2x 1
1
b
9
4
4b 3
784a4 =19 a19 = 79 n = (19 - 4) + 1 = 16
)98(8)7919(2
1616 S
)62(5.8)4715(2
1717 S
a7 =15 a23 = 47 n = (23-7) + 1 = 17
53
19, 23, 27, 31…79
15, 17, 19, …47
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An geometric series is a series associated with a geometric sequence. They can be
infinite or finite. Finite and infinite have
different formulas depending on the
value of r.54
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1, 4, 7, 10, 13, …. Infinite Arithmetic(constantly getting larger or smaller)
3, 7, 11, …, 51 Finite Arithmetic n 1 nnS a a2
1, 2, 4, …, 64 Finite Geometric
1, 2, 4, 8, …Infinite Geometricr < -1 OR r > 1
(constantly getting larger or smaller)
“diverges”
Infinite Geometric-1 < r < 1
“converges”
No Sum
No Sum
rraS
n
n
1
)1(1
55r
aS
1
1...
271,
91,
31,1,3
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71 1 1Find S of ...2 4 8
11184r
1 1 22 4
?
721
1)1(
71:
11
rn
arraS
termsstofsumFindFiniten
n
211
))21(1(
21 7
7
S
128127
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Sums of Infinite Series Made Finite
(referred to as partial sums)
Infinite SeriesFinding the Sum
of Infinite Sequences“Converges” vs. “Diverges”
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Find the sum, if possible:
1 1 11 ...2 4 8
1 1
12 4r11 22
Is -1 < r < 1? Yes (Infinite Series - converges)
59
raS
1
1
211
1
S 2
211
Geometric~need to find r~
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Find the sum, if possible:
2 2 8 16 2 ... 8 16 2r 2 2
82 2
NO SUM Is -1 < r < 1? No (Infinite series - Diverges)
60
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Find the sum, if possible:
Is -1 < r < 1? Yes (Infinite Series – Converges)
61
raS
1
1
321
1
S 3
...278
94
321
32
3294
132
r
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Find the sum, if possible:
2 4 8 ...7 7 7
4 87 7r 22 47 7
NO SUM
Is -1 < r < 1? No (Infinite Series–Diverges)
62
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Find the sum, if possible: 510 5 ...2
5
5 12r10 5 2
-1 < r < 1 Yes (Infinite Series–Converges)
63
raS
1
1
211
10
S 20
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0
n
b
365
036
5
1365
2365
...
1aS1 r
6 15315
2
x
3
7
2x 1
2 1 2 8 1 2 9 1 ...7 2 123
n 1 n2n 1S a a 15
23
27 47
527
47...,19,17,15
64
Finding the Sum from Sigma Notation
53
r so “converges”
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Rewrite using sigma notation: 3 + 6 + 9 + 12
Arithmetic, d= 3
n 1a a n 1 d
na 3 n 1 3
Explicit formula
65
4th term
4
1st term
n=1
na 3n n3
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Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1
Geometric, r = ½ n 1
n 1a a r
66
Explicit formula
n 1
n1a 162
1
2116
n
n=1
1st term
5
5th term
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Rewrite the following using sigma notation:3 9 27 ...5 10 15
Numerator is geometric, r = 3Denominator is arithmetic d= 5
NUMERATOR: n 1n3 9 27 ... a 3 3
DENOMINATOR: n n5 10 15 ... a 5 n 1 5 a 5n
SIGMA NOTATION: 1
1
n
n 5n3 3
67