Linear Sequences & nth Terms

Identifying Linear Sequences

A linear sequence is a list of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference.

Such sequences are also known as arithmetic progressions.

For example, the sequence:

3, 7, 11, 15, 19, ...

has a common difference of 4 because each term increases by 4 from the previous one.

If the difference between terms is not constant, the sequence is not linear (e.g., quadratic or cubic sequences, which are covered separately).

To recognise a linear sequence:

• Check the difference between consecutive terms.
• If the difference is the same throughout, the sequence is linear.
• Examples of linear sequences include:

• 5, 8, 11, 14, 17, ... (common difference 3)
• 20, 18, 16, 14, 12, ... (common difference -2)
• 0, 5, 10, 15, 20, ... (common difference 5)

For instance, the sequence 2, 4, 6, 8, 10 has a constant difference of 2, so it is linear.

Finding the nth Term

The nth term of a linear sequence is a formula that gives the value of any term in the sequence based on its position n.

The general form of the nth term for a linear sequence is:

$$\text{nth term} = an + b$$

where:

• a is the common difference (the amount the sequence increases or decreases by each time)
• b is a constant that adjusts the sequence to start at the correct value

To find a, subtract the first term from the second term (or any term from the previous term):

$$a = \text{difference between consecutive terms}$$

To find b, substitute the value of n and the corresponding term into the formula and solve for b.

For example, consider the sequence:

4, 7, 10, 13, 16, ...

The common difference $a = 7 - 4 = 3$.

Using the formula $an + b$, substitute $n=1$ and term = 4:

$$3 \times 1 + b = 4 \implies b = 4 - 3 = 1$$

So the nth term formula is:

$$\text{nth term} = 3n + 1$$

Using nth Term Formula

Once the nth term formula is known, it can be used to:

• Calculate any term in the sequence
• Check if a number is part of the sequence
• Generate terms of the sequence

To calculate a specific term, substitute the term number into the nth term formula.

For example, using the formula $\text{nth term} = 3n + 1$, the 10th term is:

$$3 \times 10 + 1 = 31$$

To check if a number $x$ is in the sequence, set the nth term equal to $x$ and solve for $n$. If $n$ is a positive integer, $x$ is in the sequence.

For example, is 22 in the sequence defined by $3n + 1$?

Set $3n + 1 = 22$:

$$3n = 21 \implies n = 7$$

Since $n=7$ is a positive integer, 22 is the 7th term of the sequence.

To generate terms, substitute values of $n = 1, 2, 3, ...$ into the nth term formula.

For example, for $\text{nth term} = 4n - 2$, the first five terms are:

• $n=1: 4 \times 1 - 2 = 2$
• $n=2: 4 \times 2 - 2 = 6$
• $n=3: 4 \times 3 - 2 = 10$
• $n=4: 4 \times 4 - 2 = 14$
• $n=5: 4 \times 5 - 2 = 18$

For example, using the nth term formula $2n + 3$, the 5th term is:

$$2 \times 5 + 3 = 10 + 3 = 13$$