4.9: Sequences

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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 “a_n” refers to the element at the n^th 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.