Concepts: sequences, series and patterns

2.1 What a sequence is

A sequence is a list of numbers in a definite order. The rule that produces the list can be a formula in n, such as the nth term equal to n squared, or a recursive rule that gives each term from the earlier ones. The index n runs over the natural numbers, or over a finite set when the sequence is finite.

The order is part of the definition. The sequences 1, 2, 3 and 3, 2, 1 are different even though they use the same numbers, because the positions differ. Two sequences are equal only when every corresponding term matches.

Reading a term from a formula is direct substitution. For the nth term equal to n(n plus 2), the first five terms come from putting n equal to 1, 2, 3, 4 and 5: 3, 8, 15, 24 and 35. For the nth term equal to 2 to the n, the terms are 2, 4, 8, 16 and 32. Checking the first few terms against the formula is the quickest way to catch an algebra slip.

A finite sequence stops after a last index, while an infinite sequence keeps going with no final term. The sum of a finite sequence is a number, and the sum of an infinite sequence may or may not settle at a value; this chapter only needs finite sums. When a question gives a formula and a term value and asks for the position, solve the resulting equation for n and keep only whole number answers, since positions are natural numbers. For example, the even numbers form an infinite sequence while the divisors of 60 form a finite one, and the difference matters because only the finite case has a last term you can name.

2.2 Recursive definitions

A recursive sequence gives the first term or terms and then states how each later term is built from the earlier ones. The Fibonacci sequence is the standard example: both starting terms are 1, and every term after that is the sum of the two before it, giving 1, 1, 2, 3, 5, 8 and so on.

Recursive rules are common in word problems. A plant grows 3 cm in the first week and then 2 cm more than the previous week; a salary starts at a value and rises by a fixed amount monthly; a machine loses a fixed percentage of its value each year. Each of these gives a rule of the form next term equals a function of the previous term, and the job is to identify the pattern and compute the required term.

A useful check is to generate a few terms both from the recursive rule and from any closed formula you guess. If they agree, the formula is probably right, though only a proof shows it always holds.

2.3 Series and sums

The series associated with a sequence is the expression formed by adding its terms. The series of the first n terms is written with the capital S and a subscript n, and it is the number obtained after the addition is completed. A series is finite or infinite together with its sequence.

The relation between consecutive sums is used constantly. The nth term is the sum of the first n terms minus the sum of the first n minus 1 terms. This simple difference turns a question about a sum into a question about a term and back again, and it appears in both the exercise set and the previous year questions.

One caution about words: a series names the indicated addition, while its value is the total you get after adding. Writing the series with dots means the pattern continues; writing S subscript n means the completed total of n terms. Keep the two words apart when writing an answer.

2.4 Exercise 8.1 patterns

The exercise asks for the first five terms from given nth-term formulas, then for indicated later terms, then for recursive sequences and their series. A typical item gives the nth term as n(n plus 2) and asks for the first five terms; another gives the rule a1 equal to 3 and a(n) equal to 3a(n minus 1) plus 2, where each term is built from the previous one.

The Fibonacci question asks for the ratio of consecutive terms for the first five positions. The ratios are 1, 2, 1.5, 1.666 and 1.6, and they approach the golden ratio, which is a useful pattern to notice but not something the chapter requires you to prove.

Questions of the type where the nth term contains (minus 1) to the n produce alternating signs. Write the first few terms with the signs worked out explicitly, and the pattern becomes clear.