Mathematics · Ch 8 — Sequences and Series
Introduction
Introduction
The Idea of a Sequence
In everyday language, a "sequence" simply means a list of things in a particular order — first, second, third, and so on. Mathematics uses the word in exactly the same way. A sequence is an ordered list of objects (usually numbers), where the order matters because each position tells you something specific.
For example, the population of human beings (or of a bacteria colony) measured at different points in time forms a sequence. The amount of money in a bank account after each year of deposits also forms a sequence. The depreciated value of a piece of equipment after each year of use is another sequence. Sequences appear naturally whenever we track something over time or in a fixed order, and they have important applications across many areas of human activity.
A sequence is not the same as a set. In a set, order does not matter — is the same as . In a sequence, order is everything: the first term, second term, third term, etc., are distinct positions.
Progressions
A sequence that follows a specific, recognisable pattern is called a progression. You have already studied one important type of progression in an earlier class: the arithmetic progression (A.P.), where each term is obtained by adding a fixed number to the previous term.
Not every sequence is a progression — a list of numbers with no consistent rule connecting one term to the next (say, an arbitrary jumble of values) is a sequence but not a progression, because there is nothing predictable to describe it by. The precise idea of a term, a general term, and how a sequence can be described by a formula or a rule is developed in the next section.
What This Chapter Covers
Building on what you already know about arithmetic progressions, this chapter goes further and studies:
- Arithmetic mean — the average of two numbers, and how to insert numbers between them so that the whole set forms an A.P.
- Geometric mean — the middle term of a geometric progression, and how it relates to the arithmetic mean.
- The relationship between A.M. and G.M. — a fundamental inequality connecting the two.
- Special series — the sum to terms of the natural numbers, the sum to terms of their squares, and the sum to terms of their cubes.
All progressions are sequences, but not all sequences are progressions. A progression must have a definite, repeatable rule that generates each term from the term(s) before it — that is what makes it possible to describe the whole list compactly instead of listing every term by hand.