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Algebra II Honors—Day 69. Warmup. Solve and check: Find the next four terms of this arithmetic sequence: 207, 194, 181, . . . Find the indicated term of this arithmetic sequence:. Warmup. Check both solutions!!!. -13. -13. -13. -13. Warmup. - PowerPoint PPT Presentation
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Algebra II Honors—Day 69
Warmup
• Solve and check:
• Find the next four terms of this arithmetic sequence:207, 194, 181, . . .
• Find the indicated term of this arithmetic sequence:
1119 xx
21,8,801 nda
Warmup
4or 3
0)4)(3(
0127
12119
)1(119
1119
2
2
2
xx
xx
xx
xxx
xx
xx
416
41127
1)3(11)3(9
Check both solutions!!!
525
51136
1)4(11)4(9
Warmup
• Find the next four terms of this arithmetic sequence:207, 194, 181, . . .
What is the common difference?? (194-207=-13 and 181-194=-13, so d=-13)
207, 194, 181, 168, 155, 142, 129
-13 -13 -13 -13
Warmup
• Find the indicated term of this arithmetic sequence:
21,8,801 nda
dnaan )1(1
8016080
)8)(20(80
)8)(121(80
21
21
21
a
a
a
Today
• Finish up Arithmetic Sequences and Series
• Essential Question/New Material
• Homework
Essential Questions
• What is an arithmetic series and how is it calculated?
• What is a geometric sequence?
• What is a geometric series, and how is it calculated?
B
nn A
a
UPPER BOUND(NUMBER)
LOWER BOUND(NUMBER)
SIGMA(SUM OF TERMS) NTH TERM
(SEQUENCE)
Sigma Notation
n (Index of
summation)
An arithmetic series is a series associated with an arithmetic sequence.
The sum of the first n terms:
12n n
nS a a
1(2 ( 1) )2n
nS a n d
OR
Find the sum of the first 100 natural numbers.
1 + 2 + 3 + 4 + … + 100
12n n
nS a a 1 1a
100na 100n
100
100(1 100)
2S
5050
Find the sum of the first 14 terms of the arithmetic series 2 + 5 + 8 + 11 + 14 + 17 +…
1 2a 3d 14n
1(2 ( 1) )2n
nS a n d
14
14(2(2) (14 1)3)
2S
14 7(3 13(3))S 7(43)301
j
4
1
j 2
21 2 2 3 2 24 18
7
4a
2a 42 2 5 2 6 72 44
n
n 0
4
0.5 2
00.5 2 10.5 2 20.5 2 30.5 2 40.5 2
33.5
Find the sum of the series
13
1
(4 5)n
n
9 13 17 ....
1 9a 4d 13n 1(2 ( 1) )
2n
nS a n d
13
13(2(9) (13 1)4)
2S
13(66)
2 429
12.2 – Geometric Sequences and Series
1, 4, 7,10,13
9,1, 7, 15
6.2, 6.6, 7, 7.4
, 3, 6
Arithmetic Sequences
ADDTo get next term
2, 4, 8,16, 32
9, 3,1, 1/ 3
1,1/ 4,1/16,1/ 64
, 2.5 , 6.25
Geometric Sequences
MULTIPLYTo get next term
Arithmetic Series
Sum of Terms
35
12
27.2
3 9
Geometric Series
Sum of Terms
62
20 / 3
85 / 64
9.75
Vocabulary of Sequences (Universal)
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
n 1
n 1
n1
n
nth term of geometric sequence
sum of n terms of geometric sequ
a a r
a r 1S
r 1ence
Recursive Formula:
an+1=an(r)
ExplicitFormula
Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic
3 9 / 2 31.5 geometric r
2 3 2
3 3 3 3 3 3
2 2 2
92, 3, , , ,
2
9 9 9
2 2 2 2 2 2
92, 3, , ,
27 81 243
4 8,
2 16
1 9
1 2If a , r , find a .
2 3
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/2
x
9
NA
2/3
n 1n 1a a r
9 11 2
x2 3
8
8
2x
2 3
7
8
2
3 128
6561
Find two geometric means between –2 and 54
-2, ____, ____, 54
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
-2
54
4
NA
x
n 1n 1a a r
1454 2 x
327 x 3 x
The two geometric means are 6 and -18, since –2, 6, -18, 54
forms a geometric sequence
2 4 1
2Find a a if a 3 and r
3
-3, ____, ____, ____
2Since r ...
3
4 83, 2, ,
3 9
2 4
8 10a a 2
9 9
9Find a of 2, 2, 2 2,...
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
x
9
NA
2
2 2 2r 2
22
n 1n 1a a r
9 1
x 2 2
8
x 2 2
x 16 2
5 2If a 32 2 and r 2, find a
____, , ____,________ ,32 2
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
x
5
NA
32 2
2n 1
n 1a a r
5 1
32 2 x 2
4
32 2 x 2
32 2 x4
8 2 x
*** Insert one geometric mean between ¼ and 4***
*** denotes trick question
1,____,4
4
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/4
3
NA
4
xn 1
n 1a a r
3 114
4r 2r
14
4 216 r 4 r
1,1, 4
4
1, 1, 4
4
7
1 1 1Find S of ...
2 4 8
1a First term
na nth term
nS sum of n terms
n number of terms
r common ratio
1/2
7
x
NA
11184r
1 1 22 4
n1
n
a r 1S
r 1
71 12 2
x12
1
1
71 12 2
12
1
128
127
Section 12.3 – Infinite Series
1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum
3, 7, 11, …, 51 Finite Arithmetic n 1 n
nS a a
2
1, 2, 4, …, 64 Finite Geometric n
1
n
a r 1S
r 1
1, 2, 4, 8, … Infinite Geometricr > 1r < -1
No Sum
1 1 13,1, , , ...
3 9 27Infinite Geometric
-1 < r < 11a
S1 r
Sequence Type Series
Find the sum, if possible: 1 1 1
1 ...2 4 8
1 112 4r
11 22
1 r 1 Yes
1a 1S 2
11 r 12
Find the sum, if possible: 2 2 8 16 2 ...
8 16 2r 2 2
82 2 1 r 1 No
NO SUM
Homework
• Handout on Arithmetic and Geometric Sequences/Series—Due Tuesday for a grade
