Transcript 13.2 Arithmetic & Geometric Sequences Today’s Date: 5/1/14 Arithmetic sequence (defined recursively) A sequence a1, a2, a3, … if there is a constant d for which.

13.2 Arithmetic & Geometric
Sequences
Today’s Date:
5/1/14
Arithmetic sequence
(defined recursively) A sequence a1, a2, a3, … if there is a constant d for
which an = an–1 + d for n > 1
(defined explicitly) the general term is an = a1 + (n – 1)d
d is the common difference d = an – an–1
Ex 1) Determine if the sequence is arithmetic. If yes, name the common
difference.
a) 20, 12, 4, –4, –12, …
yes
d = –8
b) 9.3, 9.9, 10.5, 11.1, 11.7, …
yes
d = 0.6
Ex 2) Create your own arithmetic sequence with common
difference of –1.5 (share a few together)
Ex 3) Find the 102nd term of the sequence 5, 13, 21, 29, …
a1 = 5
a102 = 5 + (102 – 1)(8)
d=8
= 5 + 808 = 813
The graph of a sequence is a set of points – NOT a continuous curve
sequence
continuous
function
If we know two terms of a sequence, we can find a formula.
Ex 4) In an arithmetic sequence, a5 = 24 and a9 = 40. Find the explicit
formula.
(write what we know)
24 = a1 + (5 – 1)d
40 = a1 + (9 – 1)d
Solve the system:
a1 + 4d = 24
– a1 +– 8d =–40
– 4d = –16
a1 + 16 = 24
d=4
a1 = 8
an = 8 + (n – 1)(4)
or an = 4 + 4n
If a1, a2, a3, …, ak–1, ak is an arithmetic sequence, then a2, a3, …, ak–1
are arithmetic means between a1 and ak.
Ex 5) Find 3 arithmetic means between 9 and 29.
14 , ___
19 , ___
24 , 29
9, ___
29  9
d
4
*this is a stream-lined way to solve*
last – first
# of commas
20
d
5
4
Geometric Sequence
(defined recursively) A sequence a1, a2, a3, …if there is a constant r for
which an = an–1 · r for n > 1
(defined explicitly) the general term is an = a1rn–1
an
r is the common ratio
r
an1
Ex 6) Create your own geometric sequence with common ratio r = –2.
(share please)
Ex 7) Find an explicit formula for the geometric sequence
4, 20, 100, 500, … and use it to find the ninth term.
a1 = 4
r
20
5
4
an  4  5n1
a9  4  591  1,562,500
If a1, a2, a3, …, ak–1, ak is a geometric sequence, then a2, a3, …, ak–1 are
called geometric means between a1 and ak.
Ex 8) Locate 3 geometric means between 4 and 324.
or
4, ___
___ , 324
12 , ___
36 , 108
4, –12
___ , ___
___ , 324
36 , –108
# of
commas
last
first
4
*stream-lined way to solve*
324 4
 81  3
4
A single geometric mean is called the geometric mean.
(the signs must be the same)
m   ab
Ex 9) Find the mean proportional m (if it exists) between:
a) –42 and –378
b) 1 and –16
DNE
m   (42)(378)  126
Homework
#1302 Pg 687 #1–3, 11, 13, 14, 15, 18, 20, 23–25, 28–30,
32, 34, 35, 38
*There are several word problems in the homework –
just make the sequence and apply the rules!