Mathematics · Ch 8 — Sequences and Series

Sequences

8.2

Sequences

What is a Sequence?

A sequence is simply an ordered list of numbers. The order matters — the first number, second number, third number, and so on, are all distinct positions. For example, the number of ancestors a person has over 300 years, assuming a 30-year generation gap, gives the sequence: 2, 4, 8, 16, 32, …, 1024. Here, the first term is 2 (parents), the second is 4 (grandparents), and so on up to the tenth term, 1024.

Another example: when you divide 10 by 3, the successive quotients you get at each step of the division are 3, 3.3, 3.33, 3.333, … This is also a sequence — it never ends, because the division never terminates.

The individual numbers in a sequence are called its terms. We denote the first term by $a_1$ , the second by $a_2$ , the third by $a_3$ , and in general, the $n$ th term (the number at the $n$ th position) by $a_n$ . The $n$ th term is also called the general term of the sequence.

In the ancestors example:

a_1 = 2,\quad a_2 = 4,\quad a_3 = 8,\quad \ldots,\quad a_{10} = 1024

In the division example:

a_1 = 3,\quad a_2 = 3.3,\quad a_3 = 3.33,\quad \ldots,\quad a_6 = 3.33333,\quad \ldots

Finite and Infinite Sequences

A sequence that contains a fixed, finite number of terms is called a finite sequence. The ancestors sequence is finite because it has exactly 10 terms.

A sequence that is not finite is called an infinite sequence. The sequence of successive quotients from dividing 10 by 3 is infinite — it goes on forever.

Note

The distinction is purely about the number of terms. A finite sequence has a last term; an infinite sequence does not.

Expressing a Sequence by a Formula

Often, we can find a rule or formula that gives the $n$ th term directly. For instance, consider the sequence of even natural numbers:

2,\ 4,\ 6,\ 8,\ \ldots

We can see:

a_1 = 2 = 2 \times 1

a_2 = 4 = 2 \times 2

a_3 = 6 = 2 \times 3

a_4 = 8 = 2 \times 4

and so on. In general, the $n$ th term is:

a_n = 2n

where $n$ is a natural number.

Similarly, the sequence of odd natural numbers:

1,\ 3,\ 5,\ 7,\ \ldots

has the $n$ th term given by:

a_n = 2n - 1

Sequences Without a Simple Formula

Not every sequence can be expressed by a neat algebraic formula. Consider the Fibonacci sequence:

1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ \ldots

At first glance, there is no obvious pattern. However, this sequence is generated by a recurrence relation — a rule that tells you how to get the next term from the previous ones. For the Fibonacci sequence:

a_1 = 1,\quad a_2 = 1

a_3 = a_1 + a_2 = 1 + 1 = 2

a_4 = a_2 + a_3 = 1 + 2 = 3

and in general, for $n > 2$ :

a_n = a_{n-2} + a_{n-1}

Another example is the sequence of prime numbers:

2,\ 3,\ 5,\ 7,\ 11,\ 13,\ \ldots

There is no known formula that gives the $n$ th prime directly. Such a sequence can only be described verbally. …

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