Common Mistakes with Quadratic Sequences in GCSE Maths

A quadratic sequence question can feel strangely unforgiving. You calculate the differences, recognise the pattern and write an expression that looks convincing. Then the mark scheme disagrees.

The common mistakes with quadratic sequences in GCSE Maths usually come from a handful of small errors: stopping at the first differences, forgetting to halve the second difference, or finding the $n^2$ term but not the complete nth term. The solution is a method that separates the process into steps and checks the result against the original sequence.

Imagine a student reviewing a question that cost them marks. Rather than deciding they are simply “bad at sequences”, they investigate where the method changed direction. That is what we will do here. Each mistake has a cause, a correction and a worked example.

The quadratic sequences checklist

For a sequence with nth term $a{n}^2+bn+c$:

• Calculate the first differences.
• Calculate the second differences.
• Check that the second differences are constant.
• Use $2a$ as the constant second difference, so divide it by $2$ to find $a$.
• Write the values of $a{n}^2$ underneath the sequence.
• Subtract these values from the original terms.
• Find the linear nth term of the remaining sequence.
• Combine the two expressions.
• Substitute at least one value of $n$ to check your answer.

You can see the full method alongside videos and practice on the Maths Genie GCSE revision lessons.

Mistake: stopping after the first differences

Consider the sequence:

$$3,8,15,24,35$$

Its first differences are:

$$5,7,9,11$$

A common error is to see that these differences increase by $2$ and then treat $2$ as the coefficient of $n^2$. But the changing first differences do not directly give the quadratic coefficient. You need the second differences:

$$2,2,2$$

A constant second difference confirms that the sequence is quadratic. For an nth term beginning with $a{n}^2$, the second difference is $2a$. Therefore:

$$2a=2$$ $$a=1$$

The nth term begins with $n^2$, not $2n^2$.

Mistake: forgetting to divide the second difference by two

This is one of the most frequent quadratic sequences mistakes because the constant second difference feels like the obvious number to use.

Take:

$$2,9,20,35,54$$

The first differences are:

$$7,11,15,19$$

The second differences are:

$$4,4,4$$

Since the second difference equals $2a$:

$$2a=4$$ $$a=2$$

The quadratic part is therefore $2n^2$. Writing $4n^2$ would make the sequence grow much too quickly.

There is a useful reason behind the rule. The sequence generated by $n^2$ has a constant second difference of $2$. Multiplying $n^2$ by $a$ multiplies that second difference by $a$, producing $2a$.

Mistake: finding only the quadratic part

Finding $a$ is an important step, but it is not usually the final answer. Return to:

$$2,9,20,35,54$$

We found that its nth term begins with $2n^2$. The first five values of $2n^2$ are:

$$2,8,18,32,50$$

Subtract these from the original terms:

$$\begin{array}{rl}2-2& =0\\ 9-8& =1\\ 20-18& =2\\ 35-32& =3\\ 54-50& =4\end{array}$$

The remaining sequence is:

$$0,1,2,3,4$$

Its nth term is $n-1$, because substituting $n=1$ gives $0$. Combine both parts:

$$2n^2+n-1$$

Now check the third term:

$$2(3{)}^2+3-1=18+2=20$$

That matches the original sequence.

Once this subtraction method feels clear, the Maths Genie Question Bank gives you a direct way to practise spotting and correcting the same errors.

A student detective carefully checking their checking during revision

Mistake: subtracting in the wrong direction

After finding $a{n}^2$, subtract its terms from the original sequence. Reversing the subtraction changes every sign and produces the wrong linear remainder.

For the sequence above, the correct calculation is:

$$original\; term-2n^2$$

For the second term:

$$9-8=1$$

It is not:

$$8-9=-1$$

Writing the two rows vertically can prevent this. Put the original sequence on top, the $a{n}^2$ sequence underneath, and subtract downwards.

Mistake: assuming the quadratic coefficient is positive

Quadratic sequences do not always rise faster and faster. Their second difference can be negative.

Consider:

$$10,15,18,19,18$$

The first differences are:

$$5,3,1,-1$$

The second differences are:

$$-2,-2,-2$$

Therefore:

$$2a=-2$$ $$a=-1$$

The quadratic part is $-n^2$. Its values are:

$$-1,-4,-9,-16,-25$$

Subtracting these from the original sequence gives:

$$11,19,27,35,43$$

This is a linear sequence with nth term $8n+3$. The complete nth term is:

$$-n^2+8n+3$$

Check the fifth term:

$$-(5{)}^2+8(5)+3=-25+40+3=18$$

The negative sign is essential. A missing sign can undo an otherwise correct method.

Mistake: misaligning the differences

A first-difference row has one fewer value than the original sequence. The second-difference row has one fewer value again. Students sometimes add an extra difference or compare terms that are not adjacent.

For four terms $u_1,u_2,u_3,u_4$, the first differences are:

$$u_2-u_1,u_3-u_2,u_4-u_3$$

There must be exactly three of them. Slow, organised subtraction is more reliable than trying to see every pattern mentally. This matters particularly when terms are negative.

Common mistakes to catch before the exam

Before moving to another question, ask:

• Did I calculate second differences rather than stop at the first?
• Did I divide the constant second difference by $2$?
• Did I subtract $a{n}^2$ from the original terms in the correct direction?
• Did I find the remaining linear expression?
• Did I keep negative signs visible?
• Did I give an expression in $n$, rather than another term in the sequence?
• Did I test my nth term using an original term?

That final check is especially valuable. A formula can look neat and still be wrong. Substitution turns confidence into evidence.

Exam-style quadratic sequences practice

Try these without looking back at the worked examples.

Question A

Find the nth term of:

$$6,15,28,45,66$$

Question B

Find the nth term of:

$$4,11,22,37,56$$

Question C

Find the nth term of:

$$12,19,22,21,16$$

Answers

For Question A, the first differences are $9,13,17,21$ and the second difference is $4$. The answer is:

$$2n^2+3n+1$$

For Question B, the first differences are $7,11,15,19$ and the second difference is $4$. The answer is:

$$2n^2+n+1$$

For Question C, the first differences are $7,3,-1,-5$ and the second difference is $-4$. The answer is:

$$-2n^2+13n+1$$

When you mark your work, do not record only whether an answer was right or wrong. Identify the precise line where it went wrong. The GCSE exam papers and mark schemes can then help you practise applying the method in a wider exam context.

Turn the method into reliable marks

Knowing the steps is not quite the same as following them accurately when the numbers become awkward. The student in our example improves by checking one decision at a time: second differences, halving, subtraction, linear remainder and substitution.

Your turn. Start the Maths Genie quadratic sequences lessons and Question Bank and practise the method until those avoidable quadratic sequence errors stop costing you marks.