Nth Term GCSE: Sequences Made Simple

A sequence can look like a row of unrelated numbers. Under exam pressure, that row may feel like a code you were somehow expected to memorise. But almost every nth term GCSE question begins with one manageable idea: identify what changes, then connect that change to the term's position.

For a linear sequence, the difference between consecutive terms is constant. That difference gives you the number in front of $n$. One final adjustment then produces the complete nth-term rule. Term-to-term rules work differently: they tell you how to move from one term to the next.

This guide explains both methods, with exam-style practice and answers aimed at GCSE Maths resit students. Linear nth terms are part of Foundation content in major GCSE specifications, including AQA and Edexcel, so this is a valuable topic to secure.

Your sequences checklist

By the end of this lesson, you should be able to:

• recognise a linear sequence from its constant difference
• continue a sequence using a term-to-term rule
• find the position-to-term rule of a linear sequence
• generate terms from an nth-term expression
• test whether a number appears in a sequence
• explain the difference between the two types of rule

If you want the lesson, worksheets and worked solutions together, open the MathsGenie Sequences nth term revision page. AQA students can use the corresponding AQA sequences revision resources.

What is a sequence?

A sequence is an ordered list of terms. Position matters: the first term is attached to $n=1$, the second to $n=2$, and so on.

Consider:

$$5,8,11,14,\dots$$

The terms increase by $3$ each time. Because the difference is constant, this is a linear sequence, sometimes called an arithmetic sequence.

That simple observation opens two different doors. You could describe how to travel through the sequence, which gives a term-to-term rule. Or you could build a shortcut to any position, which gives a position-to-term rule.

A GCSE student discovers the nth term shortcut past a long queue of sequence terms

Term-to-term rules

A term-to-term rule tells you how to obtain the next term from the current one. For the sequence

$$5,8,11,14,\dots$$

the rule is:

$$add3$$

Starting with $5$, adding $3$ repeatedly generates the sequence.

A decreasing sequence works in the same way. For example:

$$22,18,14,10,\dots$$

The rule is:

$$subtract4$$

Term-to-term rules are useful when a question asks for the next term. Their weakness is that you must work through all the terms in between. Finding the hundredth term by repeatedly adding would be slow and vulnerable to mistakes.

Some sequences use more complicated term-to-term operations, such as multiplying by a fixed number or using the previous two terms. For sequences Foundation maths, however, being completely secure with constant increases and decreases is an important starting point.

Position-to-term rules and the nth term

An nth-term formula connects each term directly to its position. It allows you to jump to the tenth, fiftieth or hundredth term without calculating every earlier value.

For a linear sequence, the rule has the form:

$$an+b$$

Here, $a$ is the constant difference and $b$ is the adjustment needed to reach the actual sequence.

Finding the nth term step by step

Find the nth term of:

$$5,8,11,14,\dots$$

The constant difference is $3$, so begin with:

$$3n$$

The sequence generated by $3n$ is:

$$3,6,9,12,\dots$$

Compare it with the required sequence:

$$5,8,11,14,\dots$$

Every required term is $2$ greater. Therefore, the nth term is:

$$3n+2$$

Always test the answer. When $n=1$:

$$3(1)+2=5$$

When $n=2$:

$$3(2)+2=8$$

Both results match, so the rule is correct.

The MathsGenie nth term revision guide includes further notes and videos if you want to see this method modelled before attempting questions independently.

Finding an nth term when the sequence decreases

Find the nth term of:

$$17,13,9,5,\dots$$

The common difference is $-4$, so start with:

$$-4n$$

This produces:

$$-4,-8,-12,-16,\dots$$

To turn the first value, $-4$, into $17$, add $21$. The rule is therefore:

$$-4n+21$$

Check the second term:

$$-4(2)+21=13$$

A negative difference means that the coefficient of $n$ must also be negative.

Using an nth-term formula

Once a formula is given, substitute the required position for $n$.

Suppose the nth term is:

$$7n-4$$

To find the twentieth term, use $n=20$:

$$7(20)-4=140-4=136$$

You may also be asked whether a particular number occurs in the sequence. To test whether $101$ is a term, solve:

$$7n-4=101$$ $$7n=105$$ $$n=15$$

Because $15$ is a positive whole-number position, $101$ is in the sequence. It is the fifteenth term.

If solving the equation produced $n=15.5$, the number would not be in the sequence. Positions cannot fall halfway between two terms.

Exam-style practice

Try these without looking at the answers. Show the difference between consecutive terms and check any nth-term formula by substituting $n=1$.

Question A

Write the next two terms and state the term-to-term rule:

$$9,14,19,24,\dots$$

Question B

Find the nth term of:

$$6,10,14,18,\dots$$

Question C

Find the nth term of:

$$31,25,19,13,\dots$$

Question D

A sequence has nth term $5n+7$. Find its thirtieth term.

Question E

Is $92$ a term of the sequence with nth term $4n+8$? Explain your answer.

Answers

Question A: The difference is $5$, so the next terms are $29$ and $34$. The term-to-term rule is to add $5$.

Question B: The difference is $4$, so begin with $4n$. Its first term is $4$, which must increase by $2$. The nth term is:

$$4n+2$$

Question C: The difference is $-6$, so begin with $-6n$. Adding $37$ gives the required first term:

$$-6n+37$$

Question D: Substitute $n=30$:

$$5(30)+7=157$$

Question E: Solve:

$$4n+8=92$$ $$4n=84$$ $$n=21$$

Since $21$ is a positive whole number, $92$ is the twenty-first term.

For more independent practice, use the sequences exam booklet, then compare your working with the provided answers. AQA students can also browse the AQA topic exam booklets.

Common mistakes in nth term GCSE questions

Using the first term as the coefficient

In the sequence

$$7,11,15,19,\dots$$

the coefficient of $n$ is $4$, not $7$. The coefficient comes from the common difference.

Giving only the term-to-term rule

Saying “add $4$” does not answer a question asking for the nth term. One rule moves between neighbouring terms; the other calculates a term from its position.

Forgetting the adjustment

If the difference is $5$, the complete answer is not automatically $5n$. Compare the sequence generated by $5n$ with the one in the question, then find the required adjustment.

Losing the negative sign

A decreasing linear sequence has a negative common difference. For a difference of $-3$, begin with $-3n$, not $3n$.

Not checking the formula

Substitute $n=1$ and $n=2$. A check takes seconds and catches sign errors before they cost marks.

Make the method stick

Sequences become easier when practice is short, repeated and checked. First identify the difference. Then write the multiple of $n$. Finally, adjust and test. The process is more reliable than trying to spot a formula in one leap.

Use the free GCSE Maths revision hub to move from revision lessons to practice questions, mini tests, predicted papers and full past papers with mark schemes. You can also place sequences into the MathsGenie revision planner, so the topic returns before you have time to forget it.

A resit rarely changes because of one heroic revision session. It changes through small methods that become dependable. Learn the difference between term-to-term and position-to-term rules, practise the steps, check your answers, and let the nth term become one less thing to fear on exam day.