Mathematical Investigations IV
Name
Mathematical Investigations IV
Iteration Forever
HARMONIC SEQUENCES and MEANS
A sequence is harmonic if the reciprocals
of its terms form an arithmetic sequence.
1.
Show that each sequence is harmonic by verifying that the sequence of reciprocals is
arithmetic.
10 5
5
3 3 3 3
,
,...
, , 2, ,...
a. 12, 6, 4, 3 ,…
b. 10, 5,
c. 3, , ,
4 7 10 13
3 2
3

2.
Given the harmonic sequence: 1,
2 1 2 1 2 1 2
,
,
,
,
,
,
, ...
3 2 5 3 7 4 9
a.
Find the arithmetic sequence that determines this harmonic sequence.
b.
Find the 40th term of the arithmetic sequence and of the harmonic sequence.
MEANS
All of us should be familiar with averages, especially the arithmetic mean, ab , abc , etc. Here,
2
3
we'll extend the concept of a mean. For two given numbers a and b, note that each mean below is
defined to fall between the two numbers.
 
 The arithmetic mean of a sequence of n numbers may be written:
n
1
a  a  ...  an
AM    ak  1 2
n k 1
n
 The geometric mean of two positive numbers a and b is defined as
More generally,

ab.
GM  n a1  a2 ... an
1
1
 of the reciprocals:  a  b 
The harmonic mean of a and b is the reciprocal of the average
 2 
More generally,
1 n 1 
HM     
a 
n
k 1
k
1



Seq & Ser. 4.1
1
a1

1
a2
 ... 
n
1
an



1
1
Rev. S06
Mathematical Investigations IV
Name
3. Simplify the expression given for the harmonic mean of two numbers a and b to write it as a
simplified fraction with no exponents.
4.
Find the arithmetic, geometric, and harmonic means. (Show your work)
a.
12 and 16
b.
A.M.
A.M.
G.M.
G.M.
H.M.
H.M.
3, 6, and 12
8
5.
a.
Insert two parallel segments in the trapezoid as shown
and find the lengths of each. Which mean applies here?
23
b.
In a right triangle, the altitude to the hypotenuse h
is the geometric mean of the two parts of the
hypotenuse (x and y). Find x, given that y = 4 and h = 6.
Alt = h
y
x
Seq & Ser. 4.2
Rev. S06
Mathematical Investigations IV
Name
6.
7.


Find the missing terms in each sequence:
1
1
a. arithmetic: , ___ , ___ ,
2
32
c. geometric:
1
1
, ___, ___ , ___ ,
2
32
e. harmonic:
1
1
, ___ , ___ , ___ , ___ ,
7
27
b. harmonic:
1
1
, ___ , ___ ,
2
32
1
1
, ___, ___ , ___ ,
2
32
(Find another set of terms.)
d. geometric:
A geometric context: Imagine two poles, 8' and 12' tall. Wires are connected from the top of
each pole to the bottom of the other. Find the height h, above the ground, where the two wires
cross.
S
Use similar triangles to complete each of the
proportions below.
P
h
h
=
=
12
8
12
8
h h
h
Add:  
8 12
R
Q
y
x
Now solve for h.

Using a
process similar to the one above, derive a general formula for h in terms of a and b.
S
P
b
a
Q
h
x
y
R
Seq & Ser. 4.3
Rev. S06
Mathematical Investigations IV
Name
8.
9.
Sue and George form a business to plow parking areas. Because of the equipment they use,
George can plow a "typical" lot in 40 minutes, whereas Sue can accomplish the task is 30
minutes. If they each plowed half of the area, it would take a total of 35 minutes to plow the
parking area (20 minutes for George, 15 minutes for Sue). Note that this is the arithmetic mean
of their times to plow the total lot. Instead, assume that they work together (at the same time)
and work until the entire lot is plowed. How long will this take? How does this relate to the
harmonic mean?
Let's see how well you remember your geometry. Two segments with lengths a and b are
drawn on a line AB , joined at D. Circle C is drawn with diameter AB . Perpendiculars are
drawn at C and D. Use the altitude to the hypotenuse theorem and the Pythagorean theorem to
find the lengths of the following segments in terms of a and b. Identify any mean that goes
with each segment. (Arithmetic, Geometric, Harmonic)
a. CD
F
b. ED
E
G
c.
FC (& EC )
d. FD
A
a
e.
D
C
B
b
EG
f. Write an inequality between the four segments in b-e, ranking them in order from smallest to
largest and use the diagram above to explain how you know this inequality is true.
<
<
<
because
Seq & Ser. 4.4
Rev. S06