Greenwood press

ROTATIONS
113
Relative distance away from the
earth’s axis of rotation based on lati-
tudinal position. A person is half as
far from the earth’s axis of rotation
when he or she is standing at 60°
latitude because cos 60° = 1/2.

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Satellite reception

X-ray diffractometry

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SEQUENCES
Sequences are sets of numbers that often share a recursive or explicit rela-
tionship. For example, the Fibonacci sequence in the form 1, 1, 2, 3, 5, 8, 13, 21,
. . . is determined by the sum of every two previous consecutive integers in its
sequence and has many real-world applications. (See Fibonacci Sequence for
several examples.) A different pattern occurs in the terms in a geometric
sequence, where consecutive terms have a constant ratio. A geometric sequence
with an initial value equal to 4 and constant ratio of –0.5 would be 4, –2, 1, –0.5,
0.25, . . . Another type of sequence based on a constant difference between terms
is called an arithmetic sequence. An arithmetic sequence with an initial value
equal to 4 and a constant difference of –0.5 would be 4, 3.5, 3, 2.5, 2, . . .
Sequences exist in applications that have discrete and predictable patterns, such
as the value of an automobile, camera aperture, music notes, or predicting the
timing of an eruption.
Automobile value is based on its original price, depreciation rate, and age.
Since the depreciation is fairly constant for a particular model, a car’s yearly prices
can be determined using a geometric sequence. The constant ratio in this case is
0.80, since the car maintains 80 percent of its value after each year. A car selling
for $20,000 new that depreciates 20 percent each year will be worth $16,000 the
next year, and $12,800 the year after that. These values can be determined by mul-
tiplying each successive term by 0.80, or using the explicit formula for a geomet-
ric sequence, 
g
n
= g
1
r
n−1
, where 
g
n
is the value of the car after the nth year, 
g
1
is the initial value of the car during the first year, and 
r is the constant ratio. In this
case, the explicit equation for the sequence is 
g
n
= 20, 000(0.80)
n−1
. The table
on the next page represents a sample blue-book listing of the value of a vehicle for
different years based on this equation. Notice that the car loses its greatest amount
of value during the first year, since a percentage of the total value is reduced from
the original price.
Standard f-stops on cameras permit the photographer to select how much
light passes through the lens. The sequence is 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22,
32. Each of the f-stop numbers on a standard lens represents half the light of the
number before it. The consecutive f-stops are in geometric sequence with the 
common ratio 
√
2.