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Introduction to Sequences

Sequences of numbers are simply lists of numbers.

For instance, both: \[3,7,11,15,19,23, \dots \]

and
\[1,1,2,3,5,8,13, \dots \] are both sequences of numbers.

We'll often be required to predict the newt few terms of a sequence, or we'll need to find a formula for its \(n^{\text{th}}\) term.

Notation & Terminology

To refer to any term of a sequence, we use the \(u_n\) notation, where \(n\) indicates the term we're referring to.

For instance, if we're dealing with the sequence \(3,7,11,15,19,23, \dots \) we would refer to the first, second and third terms as: \[u_1 = 3\] \[u_2 = 7\] \[u_3 = 11\] We'll often refer to the \(n^{\text{th}}\) term of a sequence, \(u_n\).
The \(n^{\text{th}}\) term is the generic term of the sequence and is usually equal to some formula with \(n\), which allows us to calculate any term of the sequence.

Exercise 1

  1. Given the sequence whose first few terms are: \[2,5,8,11,14, \dots \]
    1. State the values of \(u_2\) and \(u_4\).

    2. Write the values of \(u_6\), \(u_7\) and \(u_8\).

  2. Given the sequence whose first few terms are: \[3,6,12,24,48, \dots \]
    1. State the values of \(u_3\) and \(u_5\).

    2. Write the values of \(u_6\), \(u_7\) and \(u_8\).

Answers Without Working

    1. \(u_2 = 5\) and \(u_4 = 11\)

    2. \(u_6 = 12\), \(u_7 = 20\) and \(u_8 = 23\).

    1. \(u_3 = 12\) and \(u_5 = 48\)

    2. \(u_6 = 96\), \(u_7 = 192\) and \(u_8 = 384\).

Formula for the \(n^{\text{th}}\) term

Many sequences have a formula, which allows us to calculate any term of the sequence directly.
For instance, the sequence whose first few terms are: \[3,7,11,15,19,23, \dots \] has a formula: \[u_n = 4n-1\] With this we can calculate any term of the sequence directly.
For example, we could check that the third term is indeed \(11\) by replacing every \(n\) we see in \(u_n = 4n-1\) by \(3\) and calculating: \[\begin{aligned} u_3 &= 4\times 3-1 \\ & = 12 - 1\\ u_3 & = 11 \end{aligned}\] Or we could even calculate the \(40^{\text{th}}\) term. Again all we would do is replace every \(n\) by \(40\) and calculating: \[\begin{aligned} u_{40} &= 4\times 40-1 \\ & = 160 - 1\\ u_{40} & = 159 \end{aligned}\]

Tutorial

In the following tutorial we learn about the formula for the \(n^{\text{th}}\) term of a sequence and learn how they can be used.

Exercise

  1. A sequence has formula \(u_n = 3n+2\).
    Calculate:
    1. Its first term, \(u_1\).

    2. Its eigth term, \(u_8\).

    3. Its fiftieth term, \(u_{50}\).

  2. A sequence has formula \(u_n = n^2+2n -3\).
    Calculate:
    1. Its second term, \(u_2\).

    2. Its fifth term, \(u_5\).

    3. Its tenth term, \(u_{10}\).

  3. A sequence has formula, \(u_n = 3\times 2^{n-1}\).
    Calculate:
    1. The first five terms of this sequence.

    2. The eleventh term, \(u_{11}\), of the sequence.

Answers Without Working