+ geometric sequences & series eq: how do we analyze geometric sequences & series? m2s unit...
TRANSCRIPT
+
Geometric Sequences & Series
EQ: How do we analyze geometric sequences & series?
M2S Unit 5a: Day 9
+Vocabulary
In a geometric sequence, the ratio of any term to the previous term is constant.
This common ratio is denoted by r.
Ex: Watch me as I work one…Tell whether the sequence
is geometric. Explain.
4,8,16,32,64,...
Geometric; r=2
+Tell whether the sequence is geometric. Explain.
2) 512, 128, 64, 8, ...
1) 1, -4, 16, -64, 256,...
; common ratio is -4yes
geometricnot
+VocabularyThe nth term of a geometric sequence with first term and common ratio r is...
1a 11 nna a r
Write a rule for the nth term of the geometric sequence.
1 14(2)nna
14) 4, 3a r
14( 3)nna13) 14, 2a r
+Write a rule and graph.
1 2(3)nna
15) One term of a geometric sequence is a 2.
The common ratio is 3. Write a rule for the
term and graph the sequence.
r
nth
Create a table of values for the sequence.Notice the points lie on an exponential curve.
n -1 0 1
-2/9 -2/3 -2na
+Write a rule and graph.
1 4(2) nna
16) One term of a geometric sequence is a 4.
The common ratio is 2. Write a rule for the
term and graph the sequence.
r
nth
Create a table of values for the sequence.Notice the points lie on an exponential curve.
n -1 0 1
1 2 4na
+Write a rule and graph.
1 3(3) nna
17) One term of a geometric sequence is a 3.
The common ratio is 3. Write a rule for the
term and graph the sequence.
r
nth
Create a table of values for the sequence.Notice the points lie on an exponential curve.
n -1 0 1
-1/3 -1 -3na
+Relationship between geometric sequences and exponential functionsThe common ratio (r) will always
represent the base (b) in an exponential function.
The first term will always be “a”
The exponent will always be “n-1”
The graph of a geometric sequence will always resemble part of an Exponential function.
+8. Pick the exponential function related to the given geometric sequence.Sequence: 4, 16, 64, 256, 1024, …
1) ( ) 2(16)na f x -=1) ( ) 4(2)nb f x -=1) ( ) 4(4)nc f x -=1) ( ) 4(2)nd f x -=
+9. Pick the exponential function related to the given geometric sequence.Sequence: 2, 6, 18, 54, …
1) ( ) 2(6)na f x -=1) ( ) 2(3)nb f x -=1) ( ) 2(2)nc f x -=1) ( ) 3(2)nd f x -=
+10. Pick the exponential function related to the given geometric sequence.Sequence: 90, 30, 10, 10/3, …
1) ( ) 90(3)na f x -=
) ( ) 90(3)nb f x =1
) ( ) 903
n
c f xæö÷ç= ÷ç ÷çè ø
11
) ( ) 903
n
d f x-æö÷ç= ÷ç ÷çè ø
+Let’s write the rule. Watch me as I work one.Write a rule for the nth term as an exponential function.
11) 972, -324, 108, -36, ... 1 927324 1927 3ar
11 11972 3nn n
na a ra
+Now you try.Give the exponential function that corresponds.
12) 6, 24, 96, 384, ...
1 624 46ar
11 16 4
nn nna a ra
13) 1, 6, 36, 216, 1296, ...
1 16 61ar
11 11 6nn nn
a a ra
+Vocabulary
An expression formed by adding the terms of a geometric sequence is called a geometric series.
The sum of the first n terms of a geometric series with common ratio r ≠ 1 is:
1 11 nn rS a r
+Find the sum of a geometric series.Watch me as I work one…
14) 7+(-21)+63+(-189)+...
Find the sum of the first 8 terms (by hand)
+Find the sum of a geometric series.Now you try.
15) 1+4+16+64+... Find the sum of the first 6 terms.
16) 1+9+81+729+... Find the sum of the first 10 terms.