Download The Doubling Sequence The doubling sequence is my name for the

The Doubling Sequence
The doubling sequence is my name for the sequence of powers
of 2.
~ = h1, 2, 4, 8, 16, 32, 64, 128, . . .i
D
Term in the doubling sequence form the basis for the binary
number system. Any natural number can are written as a sum of
powers of 2. For instance,
27 + 23 + 21 + 20 = 138
The Doubling Sequence
The doubling sequence is geometric. Geometric sequences have
the form
1, r, r2 , r3 , r4 , . . .
Terms in the doubling sequence lie on the exponential curve y =
2x .
32
16
8
4
0
Graph of the doubling Sequence
0
1
2
3
4
5
The Doubling Sequence in Pascal’s Triangle
Summing values in a row of Pascal’s triangle generates the doubling sequence.
0
1
2
3
4
5
0
1
1
1
1
1
1
1
1
2
3
4
5
2
1
3
6
10
3
1
4
10
4
5
1
5 1
1
Row Sum
1
2
4
8
16
32
Computing Terms in the Doubling Sequence
Terms in the Doubling sequence can be computed by the function
d ( n ) = 2n
(for all natural numbers n)
Terms in the doubling sequence can also be computed by an initial
condition
d0 = 1 (the first term is 1)
and a recurrence equation
dn = 2dn−1
(the next term equals the twice the previous term)
The Powers of Ten
The power of ten sequence is also important and interesting.
~ = h1, 10, 100, 1000, 10000, . . .i
10
The Powers of Ten web site provides interesting information
about the very small to the very large.
For back-of-the-envelope conversions a useful approximation is
210 = 1024 ≈ 1000 = 103
The Powers of Ten
Powers of 10 are given names that are frequently used in science
and engineering.
Prefix
Symbol
Power of 10
peta
tera
giga
mega
kilo
milli
micro
nano
P
T
G
M
K
m
µ
n
1015
1012
109
106
103
10−3
10−6
10−9
2