When numbers are placed consecutively and separated by commas, the result is a number sequence. A number sequence can be either finite or infinite. There are three points at the end of an infinite sequence.

The numbers that follow the marked numbers in the sequence [[$ 1, 4, 7, 10, 13,… $]] are easy to deduce, because they are formed according to a certain rule.

The number sequence can be thought of as a function with a domain of {[[$ 1, 2, 3,…, n $]]} and a range of {[[$ f (1), f (2), f (3), ... , f (n) $]]}. Thus, the terms of a number sequence can be listed by replacing the variable of the function by the sequence number. Generally, the sequence number [[$ n $]] is used as a variable in number strings.

The number sequence is usually formed according to a certain rule, in which case the sequence number of the term [[$ n $]] gives the term itself. The rule can often be interpreted as a function between positive integers [[$ 1, 2, 3, 4, ... $]] and terms in the sequence [[$ f (n) = a_n $]].

Example 1

Determine the first three terms and the 15th term of a number sequence, when the general term of the sequence is [[$ a_n = 3 + 2n $]].

[[$ n $]] describes the sequence number of a term, so the first term is obtained by placing [[$ 1 $]], etc. in place of [[$ n $]], etc.

[[$ a_1 = 3 + 2 \cdot 1 = 5 $]]​

[[$ a_2 = 3 + 2 \cdot 2 = 7 $]]​

[[$ a_3 = 3 + 2 \cdot 3 = 9 $]]​

[[$ a_{15} = 3 + 2 \cdot 15 = 33 $]]​

Example 2

What is the hundredth term of the sequence?

[[$ 1 \space 4 \space 7 \space 10 \space 13 \space ... \space ? \\ \\ {\color{blue} {\text {1. 2. 3. 4. 5. ... 100.}}} $]]​

Solution:

The next term in the sequence is obtained by adding the number three to the previous term. It would be very tedious to figure out the answer by going through all hundred terms. Indeed, a rule can be found for how the terms of a number sequence are formed. The rule utilizes the sequence number of the term. A function machine can be used for reasoning, which generates descriptions as follows:

[[$ 1 \rightarrow 1 2 \rightarrow 4 3 \rightarrow 4 4 \rightarrow 10 5 \rightarrow 13 $]]​

In general, the mode of operation can be expressed in the form [[$ n\rightarrow 3n - 2 $]]. In other words, the general term for the sequence is [[$a_n = 3n - 2$]]. The hundredth term is then [[$ 3 \cdot 100 - 2 $]].

Answer: The 100th term is [[$ 298 $]]​.