Common Mistakes with Arithmetic Sequences in GCSE Maths

Common mistakes with arithmetic sequences in GCSE Maths

You recognise the pattern. You can write the next term. Then the question asks for the nth term, and an answer that looked sensible suddenly stops working.

The most common mistakes with arithmetic sequences in GCSE maths are confusing the difference with the whole rule, counting from the wrong position, losing negative signs and accepting answers without checking them. The fix is a repeatable method: find the difference, build the rule and test it against the original sequence.

Imagine a student reviewing a practice paper. Rather than writing “revise sequences” beside every error, they identify exactly where their method went wrong. That turns a vague weakness into something they can repair.

Let's follow that approach, one mistake at a time.

Your arithmetic sequences checklist

An arithmetic sequence has a constant difference between consecutive terms. Before answering, check:

• Difference: subtract each term from the next, keeping the order consistent.
• Position: the first term normally corresponds to $n=1$.
• Rule: use $an+b$, where $a$ is the common difference.
• Verification: substitute at least two known positions.
• Membership: a term's position must be a positive whole number.

MathsGenie's Sequences (Nth Term) lesson brings the topic together with lessons, exam-style questions and worked solutions.

Mistake: using the difference as the whole rule

Consider the sequence:

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

Our hypothetical student spots that each term increases by $4$ and writes $4n$.

Why it happens: finding the difference feels like finding the pattern. But the difference tells you how quickly the sequence changes, not where it starts.

Build the correct nth term

First, calculate the common difference:

$$11-7=4$$

The rule therefore starts with $4n$. Compare its outputs with the given terms:

$$4n:4,8,12,16,\dots$$

Every required term is $3$ larger. The correct rule is:

$$4n+3$$

Now check two positions:

$$n=1:4(1)+3=7$$ $$n=3:4(3)+3=15$$

Both match. The difference gives the coefficient of $n$; the adjustment places the sequence correctly.

Mistake: adding the first term instead of the adjustment

For that same sequence, another tempting answer is $4n+7$.

Why it happens: the student remembers that the rule needs the difference and something connected to the first term. They combine the two visible numbers without checking their roles.

But substituting $n=1$ gives:

$$4(1)+7=11$$

That is the second term, not the first.

Use the first term to find the constant

Write the unknown adjustment as $b$:

$$4n+b$$

At the first position, the output must be $7$:

$$4+b=7$$ $$b=3$$

More generally, an arithmetic sequence with first term $u_1$ and common difference $d$ has the rule:

$$u_n=u_1+(n-1)d$$

There are $n-1$ steps from the first term to the term at position $n$. That explains why adding the first term directly to $dn$ usually shifts everything along.

Turn this difference-adjust-check routine into a habit with MathsGenie's Lessons and Question Bank practice, using a worked solution to investigate any mismatch.

A minimalist stick-figure comic about checking confidence instead of checking an answer

Mistake: losing the sign in a decreasing sequence

Now the student reaches:

$$18,13,8,3,\dots$$

They write $5n+13$, treating the gap as positive.

Why it happens: “the numbers are five apart” sounds correct, but it leaves out the direction of change.

Keep subtraction in the same order

Calculate next term minus previous term:

$$13-18=-5$$

The rule begins with $-5n$. Use the first term to find the adjustment:

$$-5+b=18$$ $$b=23$$

Therefore:

$$u_n=23-5n$$

Check the second term:

$$23-5(2)=13$$

Do not change a negative difference into a positive one because “difference” sounds like a distance. In sequences, its sign matters.

For another explanation of constructing and checking rules, use the OCR sequences revision guide.

Mistake: confusing a term with its position

Suppose the question asks whether $74$ belongs to the sequence with nth term $4n+3$.

The student calculates $4(74)+3$.

Why it happens: they see a number and substitute it for $n$. But $n$ represents a position, not the value you are searching for.

Set the rule equal to the proposed term

To test membership, solve:

$$4n+3=74$$ $$4n=71$$ $$n=\frac{71}{4}=17.75$$

This is not a whole-number position, so $74$ is not in the sequence.

Compare that with $75$:

$$4n+3=75$$ $$n=18$$

So $75$ is the eighteenth term.

The question determines the operation: substitute a position to find a term; solve an equation to find a position.

Mistake: assuming every sequence is arithmetic

Consider:

$$2,5,10,17,\dots$$

A student notices the first increase is $3$ and immediately starts building a rule beginning with $3n$.

Why it happens: they recognise a familiar question format and begin the method before checking whether it applies.

Calculate several consecutive differences:

$$5-2=3,10-5=5,17-10=7$$

The differences are not constant. This is not an arithmetic sequence, so a linear rule cannot describe it.

You do not need to find its actual rule to spot the mistake. You only need to recognise that the arithmetic-sequence method is unsuitable.

Common mistakes: the final checking routine

The useful change in our student's approach is not writing more. It is checking the right things.

Before moving on:

• Test $n=1$ to check the starting term.
• Test another position to check that the rule keeps working.
• Recheck negative signs in decreasing sequences.
• Read whether the question wants a term, a rule or a position.
• Show the equation when deciding whether a number belongs.

Checking two terms is particularly useful for a proposed linear rule: one matching term alone does not establish the correct difference.

Exam-style practice

Try these without looking at the answers. Write enough working to make your method visible.

• Find a rule: determine the nth term of $6,10,14,18,\dots$.
• Handle a decrease: determine the nth term of $21,15,9,3,\dots$.
• Find a term: calculate the twentieth term when $u_n=3n-2$.
• Test membership: decide whether $62$ belongs to the sequence $u_n=5n+2$.

Answers and checks

The first sequence has difference $4$, giving $4n+2$. Substituting $n=1$ returns $6$.

The decreasing sequence has difference $-6$, giving $27-6n$. Its second term is $27-6(2)=15$.

The twentieth term is:

$$3(20)-2=58$$

For membership:

$$5n+2=62\Rightarrow n=12$$

Therefore $62$ belongs to the sequence.

For a longer independent attempt, the sequences exam booklet provides topic practice with answers. When reviewing, label the cause of each error before trying a similar question again.

Turn corrected mistakes into reliable marks

Your goal is to recognise the mistake before it reaches your final answer. Understanding these corrections is the first step; applying the same checks independently is the next. Start with MathsGenie's Lessons, then use the Question Bank to practise the arithmetic-sequence errors you identified.