Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which each number is obtained by adding a constant value to the previous number. Arithmetic sequences are linear. That is, they form a straight line when plotted on a graph.

Arithmetic sequences are useful for many mathematical modelling scenarios. They may be used in finance when calculating interest payments or instalments, in algorithms for the scheduling of events at regular intervals and in construction when objects are arranged in a linear pattern.

Use this page to revise the following concepts within Arithmetic Sequence:

The common difference

In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is called the common difference and is denoted by \(d\).

If the common difference is positive, the sequence is increasing. If the common difference is negative, the sequence is decreasing.

In an arithmetic sequence, the first term is referred to as \(a\). For example, in the sequence \(1, 4, 7, 10...\) the sequence begins at \(1\) and increases by \(3\) each term. Therefore, for this sequence, \(a = 1\) and \(d = 3\).

This would be a linear graph, such as \(y=1+3(x - 1)\) and the graph would be:

A graph of y\ =\ 1+3(x-1).

Worked Example

Determine which of the following sequences are arithmetic sequences, and for those sequences which are arithmetic, state the values of \(a\) and \(d\).

Example 1

\(2, 5, 8, 11, 14, \ldots\)

To determine if this is an arithmetic sequence, we look at the difference between consecutive terms to see if it is constant.

\[\begin{align}5 - 2&= 3 \\ 8 - 5 &= 3 \\ 11 - 8 &= 3 \\ 14 - 11 &= 3\end{align}\]

The differences between consecutive numbers are constant, so the sequence is arithmetic with a common difference of 3.

\(a\) is the first number in the sequence.

Therefore, \(a = 2\) and \(d = 3\).

Example 2

\(3, 5, 9, 17, 33, \ldots\)

\[\begin{align}5 - 3 & =3 \\ 9 - 5 &= 4 \\ 17 - 8 &= 6\end{align}\]

The differences between consecutive numbers here are not constant, so the sequence is not arithmetic.

Equations representing arithmetic sequences

The first term, \(a\), and the common difference, \(d\), can be used to create a formula to determine any term of an arithmetic sequence.
The general formula for an arithmetic sequence is

\[u_{n} = a + nd\]

Where

  • \(u_n\) is the \(n^{th}\) term after the first (\(u_0\) is the first term)
  • \(a\) is the first term of the sequence
  • \(d\) is the common difference
  • \(n\) is the position of the term in the sequence

Note that the first term, \(a\), corresponds to \(n=0\), so \(u_0\). The second term is \(n = 1\), so \(u_1\), and so on.

Note

There are other ways of representing the general formula for an arithmetic sequence.

Common alternatives include

\(a_n = a_1 + \left(n-1\right)d\) where the first term is \(a_1\)
\(t_n = a + \left(n-1\right)d\) where a is the first term and corresponds to \(t_1\)
These are all essentially different ways of showing the same relation. This resource uses the \({u}_n = a + nd\) notation.

Worked Example

Determine the equations that represent the following arithmetic sequences by finding \(a\) and \(d\).

Example 1

\(3, 6, 9, 12, 15, …\)

The first term is \(3\), hence \(a = 3\)    
The common difference is found by subtracting one number from the following number in the sequence.

\[\begin{align}d &= u_{2} - u_{1} \\ &= 9 - 6 \\ &= 3\end{align}\]

Substitute the values for \(a\) and \(d\) into the formula for arithmetic sequences.

\[\begin{align}u_{n} &= a + nd \\ u_n &= 3 + 3n\end{align}\]

This can verified by testing some numbers from the sequence.

\[\begin{align}u_4 &= 3 + (3 \times 4) \\ u_4 &= 15\end{align}\]

Remember that the first number in the sequence is \(u_0\), so \(u_4\) corresponds to the fifth number in the sequence, which is \(15\).

Example 2

\(40, 33, 26, 19, 12, …\)

\[a = 40\]

\[\begin{align}d &= u_2 - u_1 \\ &= 26 - 33 \\ &= -7\end{align}\]

Substitute the values for \(a\) and \(d\) into the formula for arithmetic sequences.

\[u_n = 40 - 7n\]

To verify,

\[\begin{align}u_3 &= 40 - 21 \\ u_3 &= 19\end{align}\]

The fourth term in the sequence given is \(19\).

It’s important to read worded questions carefully to determine the correct value for \(n\). For example, compare the two examples below.

Worked Example

Example 1

A biology class is measuring the growth of a plant.

In the first week, they measure the plant as being \(5\text{ cm}\) tall. In the second week, they measure it as being \(7\text{ cm}\) tall, and in the third week, they measure the plant as being \(9\text{ cm}\) tall.

Assuming it continues growing at this constant rate, how tall is the plant after five weeks?

The initial value, \(a\), is \(5\text{ cm}\). The constant difference, \(d\), is \(2\).

Since the first week is \(n = 0\), the fifth week would be \(n = 4\).

\[\begin{align} u_{4} &= 5 + (2 \times 4) \\ u_4 &= 13\end{align}\]

After \(5\) weeks, the plant will be \(13\text{ cm}\) tall.

Example 2

A construction crew is building a new road. At the start of the day, \(1500\text{ m}\) of the road has already been built. Every hour, they are able to extend it by \(500\text{ m}\).

If they work for 8 hours, how long will the road be when the stop for the day?

The initial value, \(a\), is \(1500\text{ m}\). As the road increases \(500\text{ m}\) each hour, the constant difference \(d\) is \(500\). Therefore, the formula for finding the length of this road after \(n\) hours is

\[u_n=1500 + 500n\]

Since the state at the start of the day is \(n = 0\), then the state after one hour corresponds with \(n = 1\), the state after two hours corresponds with \(n = 2\), and so on. In this example, the length after \(8\) hours is calculated by finding the value when \(n = 8\), not \(n = 7\).

\[\begin{align}u_8 &= 1500 + (500 \times 8) \\ u_8 &= 5500\end{align}\]