Sequences

Resources

A sequence is an ordered list of numbers. It can be thought of as a function, ƒ(n ), where the argument, n, takes on the natural-number values 1, 2, 3, 4, … (or occasionally 0, 1, 2, 3, 4, …). A sequence can follow a regular pattern or an arbitrary one. It may be possible to compute the value of f(n ) with a formula, or it may not.

The terms of a sequence are often represented by letters with subscripts, a_n, for example. In such a representation, the subscript n is the argument and tells where in the sequence the term a_n falls. When the individual terms are represented in this fashion, the entire sequence can be thought of as the set, or the set where n is a natural number. This set can have a finite number of elements, or an infinite number of elements, depending on the wishes of the person who is using it.

One particularly interesting and widely studied sequence is the Fibonacci sequence: 1, 1, 2, 3, 5, 8, … . It is usually defined recursively: a_n=a_n-2 + a_n-1. In a recursive definition, each term in the sequence is defined in terms of one or more of its predecessors (recursive definitions can also be called iterative). For example, a₆ in this sequence is the sum of 3 and 5, which are the values of a₄ and a₅, respectively.

Another very common sequence is 1, 4, 9, 16, 25, …, the sequence of square numbers. This sequence can be defined with the simple formula a_n = n, or it can be defined recursively: a_n = a_n-1 + 2n - 1.

Another sequence is the sequence of prime numbers: 2, 3, 5, 7, 11, 13, … . Mathematicians have searched for centuries for a formula which would generate this sequence, but no such formula has ever been found.

One mistake that is made frequently in working with sequences is to assume that a pattern that is apparent in the first few terms must continue in subsequent terms. For example, one might think from seeing the five terms 1, 3, 5,7, 9 that the next term must be 11. It can, in fact, be any number whatsoever. The sequence can have been generated by some random process such as reading from a table of random digits, or it can have been generated by some obscure or complicated formula. For this reason, a sequence is not really pinned down unless the generating principle is stated explicitly. (Psychologists who measure a subject’s intelligence by asking him or her to figure out the next term in a sequence are really testing the subject’s ability to read the psychologist’s mind.) Sequences are used in a variety of ways. One example is to be seen in the divide-and-average method for computing square roots. In this method, one finds the square root of N by computing a sequence of approximations with the formula a_n = (a_n-1 + N/a_n-1)/2. One can start the sequence using any value for a₁ except zero (a negative value will find the negative root). For example, when N = 4 and a₁ = 1

a₁ = 1.0

a₂ = 2.5

a₃ = 2.05

a₄ = 2.0006

a₅ = 2.0000

This example illustrates several features that are often encountered in using sequences. For one, it often only the last term in the sequence that matters. Second, the terms can converge to a single number. Third, the iterative process is one that is particularly suitable for a computer program. In fact, if one were programming a computer in BASIC (Beginners All-purpose Symbolic Instruction Code), the recursive formula above would translate into a statement such as R = (R + N/R)/2.

Not all sequences converge in this way. In fact, this one does not when a negative value of N is used. Whether a convergent sequence is needed or not depends on the use to which it is put. If one is using a sequence defined recursively to compute a value of a

KEY TERMS

Convergent —A sequence is convergent if, as one goes further and further down the list, the terms from some point on get arbitrarily close to a particular number.

Divergent —A sequence that is not convergent is divergent.

Sequence —A sequence is a series of terms, in which each successive term is related to the one before it by a fixed formula.

particular number only a convergent sequence will do. For other uses a divergent sequence may be suitable.

Mortgage companies often provide their customers with a computer print-out showing the balance due after each regular payment. These balances are computed recursively with a formula such as A_n = (A_n-1)(1.0075) - P, where A_n stands for the balance due after the n-th payment. In the formula (A_n-1)(1.0075) computes the amount on a 9% mortgage after one month’s interest has been added, and (A_n-1)(1.0075) - P the amount after the payment P has been credited. The sequence would start with A₀, which would be the initial amount of the loan. On a 30-year mortgage, the size of P would be chosen to bring A₃₆₀ down to zero. This sequence converges, but very slowly for the first few years.

Tables, such as tables of logarithms, square roots, trigonometric functions, and the like are essentially paired sequences. In a table of square roots, for example

1.0

1.00000 1.1

1.04881 1.2

1.09545

the column on the left is a sequence and the columnon the right the sequence where each b_n equals the square root of a_n. By juxtaposing these two sequences, one creates a handy way of finding square roots.

Sequences are closely allied with (and sometimes confused with) series. A sequence is a list of numbers; a series is a sum. For instance 1/1, 1/2, 1/3, 1/4, … is a harmonic sequence; while 1/1 + 1/2 + 1/3 + 1/4 +… is a harmonic series.

Resources

BOOKS

Burton, David M. The History of Mathematics: An Introduction. New York: McGraw-Hill, 2007.

Finney, Ross L., et al. Calculus: Graphical, Numerical, Algebraic. Glenview, IL: Prentice Hall, 2003.

Jeffrey, Alan. Mathematics for Engineers and Scientists. Boca Raton, FL: Chapman & Hall/CRC, 2005.

Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

J. Paul Moulton