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Ch 9.2 Wkst AP Calculus BC
Name:
Geometric Series
The n th Term of a Geometric Sequence
an  a 1 r
n 1
,
The Sum of a FINITE Geometric Series
where r  1
Sn 
a 1 (1  r n )
1 r
,
where r  1
Note: in a finite geometric series, think of n as the number of terms.
Convergence Test for a Geometric Sequence
If r  1 , then
The Sum of an INFINITE Geometric Series

 an
Sn 
is a convergent series.
n 1
a1
1 r
,
where r  1
Find (a) an expression for the n th term of each geometric sequence, (b) the sum of the first 10 numbers of the sequence,
and then find (c) the sum of the infinite geometric series.
1. 100, 50, 25, 12.5, . . .
2. 3,  12 , 48  192 , . . .
a)
a)
b)
b)
c)
c)
3. 8, 12, 18, 27, . . .
4. 6,  2 ,
a)
a)
b)
b)
c)
c)
2 2
 , ...
3 9
Find the sum of each geometric series.
1 1 1 1
   ...
2 4 8 16
5.
 2 1
7.
2 (0.1) 5  2 (0.1) 6  . . .  2 (0.1)13
3 3 3
3
   ...
2 4 8
210
6.
3
8.
 (0.75) n

n0

9.

n 1
3n

10.
1
 
3
n 1  

n
20
11. Evaluate the geometric series
(A)
3 17  1
2  3 20
(B)

3 20  1

0
(B)
(C)
2  3 23
12. Evaluate the geometric series
(A)
n
1
  .
3
n4  
1
2
3 21  1
2  3 24
(D)
0
(E)
divergent
(D)
1
54
(E)
divergent
n
1
  .
3
n4  

(C)
1
18

13.

en
n 1
n 1 3
(A)

e
3e
(B)
3e
3e
(C)
e3
3
(D)
ANSWERS:
1a) 100  (0.5) n 1
b) 199.8046875
c) 200
2a) 3 ( 4) n 1
b) -629,145
c) divergent
3a) 8  (1.5) n 1
b) 906.640625
c) divergent
5) -4/3
11) A
6) 5.997
7) 2 / 90,000
12) D
13) B
4a) 6  (1 / 3) n 1
b) 4.49923792
c) 4.5
8) 4
9) divergent
10) 0.5
e3
(E)
divergent