Transcript Section 13.1

Unit 5 – Series, Sequences, and Limits
Section 5.1 – Arithmetic and Geometric Sequences
Calculator Required
An introduction…………
1, 4, 7, 10, 13
35
2, 4, 8, 16, 32
9, 1,  7,  15
6.2, 6.6, 7, 7.4
,   3,   6
12
9,  3, 1,  1/ 3
20 / 3
1, 1/ 4, 1/16, 1/ 64 85 / 64
, 2.5, 6.25
9.75
27.2
3  9
Arithmetic Sequences
ADD
To get next term
Arithmetic Series
Sum of Terms
62
Geometric Sequences
MULTIPLY
To get next term
Geometric Series
Sum of Terms
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
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
Vocabulary of Sequences (Universal)
a1  First term
an  nth term
n  number of terms
d  common difference
nth term of arithmetic sequence  an  a1  n  1 d
Given an arithmetic sequence with a15  38 and d  3, find a1.
x
a1  First term
38
an  nth term
15
n  number of terms
-3
d  common difference
an  a1  n  1 d
38  x  15  1 3 
X = 80
Try this one: Find a16 if a1  1.5 and d  0.5
1.5
a1  First term
x
an  nth term
16
n  number of terms
0.5
d  common difference
an  a1  n  1 d
a16  1.5  16  1 0.5
a16  9
Find n if an  633, a1  9, and d  24
9
a1  First term
633
an  nth term
x
n  number of terms
24
d  common difference
an  a1  n  1 d
633  9   x  1 24
633  9  24x  24
X = 27
Find d if a1  6 and a 29  20
-6
a1  First term
20
an  nth term
29
x
n  number of terms
d  common difference
an  a1  n  1 d
20  6   29  1 x
26  28x
13
x
14
Find two arithmetic means between –4 and 5
-4, ____, ____, 5
-4
a1  First term
5
an  nth term
4
n  number of terms
x
d  common 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
Find three arithmetic means between 1 and 4
1, ____, ____, ____, 4
1
a1  First term
4
an  nth term
5
n  number of terms
x
d  common difference
an  a1  n  1 d
4  1   5  1 x 
3
x
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
Vocabulary of Sequences (Geometric)
a1  First term
an  nth term
n  number of terms
r  common ratio
nth term of geometric sequence  an  a1r n1
Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic
3 9/2
3

 1.5  geometric  r 
2
3
2
9 9 3 9 3 3 9 3 3 3
2, 3, ,  ,   ,   
2 2 2 2 2 2 2 2 2 2
9 27 81 243
2, 3, ,
, ,
2 4 8 16
If a1 
1
2
, r  , find a9 .
2
3
a1  First term
1/2
an  nth term
x
n  number of terms
r  common ratio
an  a1r n1
 1  2 
x    
 2  3 
9
2/3
9 1
28
27
128
x

8 
8
23
3
6561
Find two geometric means between –2 and 54
-2, ____, ____, 54
a1  First term
-2
an  nth term
54
n  number of terms
r  common ratio
4
x
an  a1r n1
54   2  x 
4 1
27  x3
3  x
The two geometric means are 6 and -18, since –2, 6, -18, 54
forms an geometric sequence
Find a 2  a 4 if a1  3 and r 
2
3
-3, ____, ____, ____
2
Since r  ...
3
4 8
3,  2,
,
3 9
 8  10
a 2  a 4  2  


9
 9 
Find a9 of 2, 2, 2 2,...
a1  First term
2
an  nth term
x
n  number of terms
r  common ratio
9
r
an  a1r n1
 2
2  2
x 2
x
x  16 2
9 1
8
2
2

2 2
 2
2
If a5  32 2 and r   2, find a2
____, ____, ____,____,32 2
a1  First term
an  nth term
x
32 2
n  number of terms
r  common ratio
5
 2
an  a1r n1
 
2  x  2 
32 2  x  2
32
32 2  4x
8 2x
5 1
4
*** Insert one geometric mean between ¼ and 4***
1
, ____,4
4
a1  First term
1/4
an  nth term
4
n  number of terms
3
r  common ratio
x
an  a1r n1
1 3 1
1 2
4  r  4  r  16  r 2  4  r
4
4
1
, 1, 4
4
1
,  1, 4
4