Mathematical Investigations IV
S&S Unit test
Name ______________
You may use a TI-30 Calculator on this exam. Justify all your work.
1.
(2 pts each) Given the numbers 8 and 128, find their:
a) arithmetic mean
b) geometric mean
8  128
 68
2
2.
8 128  32
Consider the series 4  7  5  8  6  9 
a. (4 pts) Write the series in
62
 k (k  3)
k 4
 62  65 .
  notation.
59
or
 (k  3)(k  6)
k 1
b. (5 pts) Use appropriate formulas to evaluate the series.
59
59
k 1
k 1
 (k  3)(k  6)   (k 2  9k  18)
59
59
59
k 1
k 1
k 1
  k 2  9 k  18
59  60 119
59  60

 9
 59 18
6
2
 87, 202
c) harmonic mean
2  8 128 256

 15.0588
8  128
17
Mathematical Investigations IV
3.
S&S Unit test
(3 pts each) If a series has a first term of 16 and the sum of its first three terms is 76, find:
a. The common ratio if the series is geometric.
16  16r  16r 2  76 
16r  16r 2  60 
4r  4r 2  15 
4r 2  4r  15  0 
(2r  3)(2r  5)  0 
3
5
r  or 
2
2
b. The common difference if the series is arithmetic.
16  (16  d )  (16  2d )  76 
3d  28 
28
d
3
4.
Name ______________
(3 pts each) Write out the first four terms of:
a. The sequence rn  n1 if
if n  1
 5

rn   3
if n  2
r  2r if n  2
n2
 n 1
5, 3, 7,1
92
b. The series
 (2
k 1
2, 4, 4, 0
k
 4k )
Mathematical Investigations IV
5.
S&S Unit test
Name ______________
n
(4 pts) Let S n   ak . Suppose that Sn  n 2  7n  1 for all positive integers n.
k 1
Determine the first three terms of the sequence ak k 1 .

a1  S1  12  7 1  1  5
a1  a2  S 2  22  7  2  1  9  a2  4
a1  a2  a3  S3  32  7  3  1  11  a3  2
5, 4, 2
6.
(3 pts each) Write:
a. A recursive formula for the sequence defined by bn 
18
.
3n
if n  1
 6

bn   1
 3  bn 1 if n  1
if n  1
 7
b. An explicit formula for the sequence defined by: cn  
cn1  2 if n  1
cn  7  2(n  1) or
cn  2n  5
Mathematical Investigations IV
S&S Unit test
Name ______________
n
7.
Let S n represent the nth partial sum of the series
 4k
k 1
8
2
1
.
a. (4 pts) Use mathematical induction to prove that S n  4 
1
Base Case: (n = 1):
 4k
k 1
8
2
8
 ,
1 3
S1  4 
4
4 8
 4   , so base case holds.
2 1  1
3 3
n
Inductive Step: I Suppose that for some positive integer n,
 4k
k 1
Then,
n 1
 4k
k 1
8
2
4
for all positive integers n.
2n  1
8 
8
8 8 8
   
 2 
 1  3 15 35
4n  1  4( n  1) 2  1
4
8
 4

, by induction hypothesis
2n  1 4( n  1) 2  1
4
8
 4

2
2n  1 4(n  2n  1)  1
4
8
 4
 2
2 n  1 4 n  8n  3
4
8
 4

2n  1 (2n  1)(2n  3)
4(2n  3)
8
 4

2
(2n  1)(2n  3) 4(n  2n  1)  1
8n  12
8
 4

2
(2n  1)(2n  3) 4(n  2n  1)  1
8n  4
 4
(2n  1)(2n  3)
4(2n  1)
 4
(2n  1)(2n  3)
4
 4
(2n  3)
4
 4
(2(n  1)  1)
So Statement holds for all positive integers n.
8
2
1
 4
4
.
2n  1
Mathematical Investigations IV
S&S Unit test
Name ______________

b. (2 pts) Use the formula in part a. to determine the value of
 4k
k 1

 4k
k 1
8
2
1
 lim  S n 
n 
4 

 lim  4 

n 
2n  1 

 40
4
8
2
1
.