Continuing Sequences

Definition of Continuing Sequences

Continuing sequences means extending a sequence beyond the terms given. This involves recognising the pattern or rule that generates the sequence and using it to predict further terms.

Sequences can be numerical or algebraic, and the pattern might be based on addition, subtraction, multiplication, division, or a combination of these. Sometimes the pattern is clear from the differences between terms, or from the ratio between terms.

The key skill is to identify the rule and apply it consistently to find the next terms.

For instance, if the sequence is 3, 6, 9, 12, ..., the pattern is adding 3 each time. So the next terms are 15, 18, 21, and so on.

Finding the nth Term

The nth term of a sequence is a formula that gives the value of any term in the sequence, where $n$ is the term number.

Sometimes you are given several terms and asked to find the nth term formula by recognising the pattern.

For continuing sequences, you may use the nth term to find terms beyond those given. The nth term can be linear (e.g. $an + b$) or non-linear (e.g. involving powers of $n$, but these are covered in other topics).

To find the nth term from given terms:

• Look at the difference between terms to check if it is constant (linear sequence).
• If constant, the nth term is of the form $an + b$.
• Use the known terms to set up equations and solve for $a$ and $b$.
• Substitute values of $n$ to find specific terms.

For example, if the sequence is 4, 7, 10, 13, ..., the difference is 3, so the nth term is $3n + b$. To find $b$, substitute $n=1$: $3(1) + b = 4$, so $b = 1$. Thus, the nth term is $3n + 1$.

Note: Non-linear sequences such as quadratic sequences are studied in other topics.

Extending Sequences

Once the nth term formula is known, you can calculate any term in the sequence, including those beyond the given terms.

To extend a sequence:

• Use the nth term formula to find the next terms by substituting $n$ values greater than the last given term number.
• Check that the terms fit the pattern to ensure consistency.

If the nth term is not given, use the pattern or rule you have identified to calculate the next terms step-by-step.

For example, for the sequence 2, 5, 8, 11, ..., the pattern is adding 3 each time. The next terms are 14, 17, 20, etc.

Inline example: Find the 5th term of the sequence 1, 3, 5, 7, ...

The difference is 2, so nth term is $2n + b$. Substituting $n=1$: $2(1) + b = 1$ gives $b = -1$. So nth term is $2n - 1$. The 5th term is $2 \times 5 - 1 = 9$.