A sequence can often be defined using an equation or rule, where the term number is the input and the output is the term or term value. Typically, the term number for the first term is 1, but it does not have to be.

The explicit rule for a sequence tells us how to calculate the term value given the term number. We can write the explicit rule using subscript notation.

The sequence: -1, 4, 9, 14, ... has the explicit rule a_n=-6+5n where a_n represents the nth term, which gives:

a_1=-6+5(1)=-6+5=-1; a_1=-1
a_2=4
a_3=9
a_4=14

Finite sequence

A sequence that has a countable or limited number of terms

Example:

2, 4, 8, 16, 32 which is a_n=2^n with term numbers: \\n\in \left\{1, 2, 3, 4, 5\right\}

Infinite sequence

A sequence that has an infinite number of terms

Example:

2, 4, 8, 16, 32... which is a_n=2^n with term numbers: n\in \Z^{+}

Worked examples

Example 1

Consider the sequence:

1, 4,9, 16, 25\ldots

Identify a_1 and a_4.

Approach

The subscript tells us the term number, so this is asking for the first and fourth terms.

Solution

The first term, a_1 of the sequence is 1.

The fourth term, a_4 of the sequence is 16.

Reflection

We should assume that the first term has a term number of 1 unless we are explicitly told otherwise.

Find the next two terms in the sequence.

Approach

We want to identify the pattern and then use it to continue the sequence.

Solution

It can be observed that the terms in the sequence are perfect squares. The first term is 1^2=1, the 2nd term is 2^2=4 and the 5th term is 5^2=25. The next two terms, a_{6} and a_{7} should be 6^2=36 and 7^2=49 respectively.

The next two terms of the sequence are 36 and 49.

Write an explicit rule for the nth term of the sequence.

Solution

Because the terms are perfect squares, the nth term of the sequence can be found by raising each term to the 2nd power. The explicit formula for the sequence is

a_n= n^2

Reflection

We can check that our rule is correct by substituting in the first few values and making sure it matches the sequence.

n	a_n
1	a_1=1^2=1
2	a_1=2^2=4
3	a_3=3^2=9

Example 2

Find the first four terms of the sequence described by:

a_n = \dfrac{3 n - 1}{n^{2} + 4}

Solution

For a sequence, the first term a_1 is when n=1. The second term a_2 is when n=2, and so on.

The value of each term is determined by substituting n into the formula.

\displaystyle a_1	\displaystyle =	\displaystyle \dfrac{3 (1) - 1}{(1)^{2} + 4}	a_1=\dfrac{2}{5}
\displaystyle a_2	\displaystyle =	\displaystyle \dfrac{3 (2) - 1}{(2)^{2} + 4}	a_2=\dfrac{5}{8}
\displaystyle a_3	\displaystyle =	\displaystyle \dfrac{3 (3) - 1}{(3)^{2} + 4}	a_3=\dfrac{8}{13}
\displaystyle a_4	\displaystyle =	\displaystyle \dfrac{3 (4) - 1}{(4)^{2} + 4}	a_4=\dfrac{11}{20}

The first four terms of the sequence are \dfrac{2}{5}, \dfrac{5}{8}, \dfrac{8}{13}, and \dfrac{11}{20}.

Outcomes

MA.912.AR.10.1

Given a mathematical or real-world context, write and solve problems involving arithmetic sequences.

MA.912.AR.10.2

Given a mathematical or real-world context, write and solve problems involving geometric sequences.

Honors: 7.01 Introduction to sequence notation

Concept summary

Worked examples

Example 1

Approach

Solution

Reflection

Approach

Solution

Solution

Reflection

Example 2

Solution

Outcomes

MA.912.AR.10.1

MA.912.AR.10.2

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