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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 Presentation
Arithmetic Sequences
1
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.
2
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
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
4
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
5
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
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
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
8
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
9
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
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:
11
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
12
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
13
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
1 29Find d if a 6 and a 20
120 6 29 x
26 28x13x14
15
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
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
17
Explicit vs. Recursive Formulas
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
Recursive Formula – (includes a1) used to find the next term of the sequence by adding the common difference to the previous term.
19
Explicit vs. Recursive Formulas
an = an-1 + 2 a1 = 4Ex: 4, 6, 8, 10…
an = an-1 + da1 = ___
an+1 = an + d a1 = ___
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
20
21
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
22
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
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
23
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
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
25
Geometric Sequences
26
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
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
28
23
23
232
4th term:
29
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?
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
30
2 4 12Find a a if a 3 and r3
2Since r ...3
2 48 10a a 2
9 9
31
-3, ____, ____, ____2 34
98
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
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
Ex: 4, 12, 36, 108…
Use a1 and r in sequence formula:
34
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
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
35
36
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
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
37
38
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
39
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
Ex: Find two geometric means between –2 and 54
-2, ____, ____, 54
n 1n 1a a r 1454 2 x
327 x 3 x
40
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
*** 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
41
Series42
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
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.
44
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
B
nn A
a
UPPER BOUNDTERM NUMBER
LOWER BOUNDTERM NUMBER
SIGMA(SUM OF TERMS)
NTH TERMSEQUENCE
(EXPLICIT FORMULA)
45
# of Terms: B – A + 1
j
4
1
j 2
21 2 2 3 2 24
18
7
4a
2a
42 2 5 2 6 72
4446
An arithmetic series is a series associated
with an arithmetic sequence.
It can be infinite or finite.
47
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
48
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
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
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
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
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
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
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
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
56
Sums of Infinite Series Made Finite
(referred to as partial sums)
Infinite SeriesFinding the Sum
of Infinite Sequences“Converges” vs. “Diverges”
57
58
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~
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
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
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
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
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”
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
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
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