Sequences
Arithmetic sequences
An arithmetic sequence is a series of
numbers obtained by adding (or subtracting) the same value
with each step. A child's counting sequence (1, 2, 3, 4, . .
.) is a simple arithmetic sequence, where the common
difference is 1: that is, each adjacent number in the
sequence differs by a value of one. An arithmetic sequence
counting only even numbers (2, 4, 6, 8, . . .) or only odd
numbers (1, 3, 5, 7, 9, . . .) would have a common
difference of 2.
In the standard notation of sequences, a
lower-case letter "a" represents an element (a single
number) in the sequence. The term "an" refers to
the element at the nth step in the sequence. For
example, "a3" in an even-counting (common
difference = 2) arithmetic sequence starting at 2 would be
the number 6, "a" representing 4 and "a1"
representing the starting point of the sequence (given in
this example as 2).
A capital letter "A" represents the sum
of an arithmetic sequence. For instance, in the same
even-counting sequence starting at 2, A4 is equal
to the sum of all elements from a1 through a4,
which of course would be 2 + 4 + 6 + 8, or 20.
Geometric sequences
A geometric sequence, on the other
hand, is a series of numbers obtained by multiplying (or
dividing) by the same value with each step. A binary
place-weight sequence (1, 2, 4, 8, 16, 32, 64, . . .) is a
simple geometric sequence, where the common ratio is
2: that is, each adjacent number in the sequence differs by
a factor of two.
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