Mastering Sequences: The Nth Term of Linear and Quadratic Sequences

Understanding sequences is a fundamental skill in GCSE maths. A sequence is simply an ordered list of numbers that follow a specific rule. By finding the 'nth term', you create a formula that allows you to calculate any number in that sequence, regardless of how far along it is. This is a powerful tool for predicting patterns and solving algebraic problems.

In this guide, we will break down the two most common types of sequences you will encounter: linear and quadratic. Mastering these will not only help you secure marks in your exams but also build a strong foundation for further study in mathematics.

Understanding Linear Sequences

A linear sequence is one where the difference between consecutive terms is constant. This is often called the 'common difference'. Because the gap never changes, the formula for the nth term will always be in the form $dn + c$, where $d$ is the common difference and $c$ is a constant.

How to find the nth term of a linear sequence

1. Find the common difference ($d$). This becomes the coefficient of $n$.
2. Write out the first few terms of the $dn$ sequence.
3. Compare your $dn$ sequence to the original sequence to find the difference ($c$).

Example 1: Find the nth term of the sequence: 5, 8, 11, 14, ...

• The difference between terms is 3. So, the formula starts with $3n$.
• The $3n$ sequence is: 3, 6, 9, 12, ...
• Compare this to our sequence (5, 8, 11, 14). We need to add 2 to each term of the $3n$ sequence to match our original.
• Therefore, the nth term is $3n + 2$.

Identifying Quadratic Sequences

A quadratic sequence is more complex. Unlike linear sequences, the first differences are not constant. However, if you calculate the 'difference of the differences' (the second difference), you will find it is constant. The general form for a quadratic nth term is $an^2 + bn + c$.

Finding the Nth Term of a Quadratic Sequence

To find the formula $an^2 + bn + c$, we follow a systematic approach. The key is to determine the values of $a$, $b$, and $c$ one by one.

Step-by-Step Method

1. Find the first differences and then the second difference.
2. The value of $a$ is always half of the second difference ($a = \text{second difference} \div 2$).
3. Subtract the $an^2$ sequence from your original sequence to leave a linear sequence.
4. Find the nth term of that remaining linear sequence to identify $bn + c$.

Example 2: Find the nth term of the sequence: 3, 8, 17, 30, 47, ...

• Terms: 3, 8, 17, 30, 47
• First differences: 5, 9, 13, 17
• Second differences: 4, 4, 4
• Since the second difference is 4, $a = 4 \div 2 = 2$. Our formula starts with $2n^2$.
• Calculate $2n^2$ for $n=1, 2, 3, 4, 5$: 2, 8, 18, 32, 50.
• Subtract $2n^2$ from the original sequence: (3-2), (8-8), (17-18), (30-32), (47-50) = 1, 0, -1, -2, -3.
• This is a linear sequence with a difference of -1. The formula is $-1n + 2$ (or $2 - n$).
• Combine them: $2n^2 - n + 2$.

Common Mistakes to Avoid

• Forgetting to square: When calculating $an^2$, students often multiply $n$ by $a$ before squaring. Always square $n$ first.
• Sign errors: When finding the linear part ($bn + c$), be very careful with negative numbers. If the sequence is decreasing, your $b$ value will be negative.
• Miscalculating the second difference: Always double-check your subtraction when finding the differences between terms. A small arithmetic error at the start will ruin the entire formula.

Frequently Asked Questions

Is the quadratic nth term only for Higher Tier? Yes, finding the nth term of a quadratic sequence is typically restricted to the Higher Tier GCSE maths papers.

What if the second difference is not constant? If the second difference is not constant, the sequence is not quadratic. It might be cubic or another type of sequence, which is generally outside the scope of standard GCSE requirements.

How do I check if my formula is correct? Substitute $n=1, 2, 3$ into your formula. If the results match the first three terms of your original sequence, your formula is correct.

Conclusion

Sequences are a vital part of your algebraic toolkit. By mastering the difference between linear and quadratic patterns, you can approach these questions with confidence. If you want to see these steps in action, head over to MathInstructor AI to generate a free, narrated animated lesson on this exact topic to visualise the patterns in motion.