Finding the nth Term of Linear Sequences Revision Notes | GCSE AQA Higher Maths

1. What Are Linear Sequences?

You see patterns everywhere. Your phone battery drops by roughly the same amount each hour. Train tickets cost £2 more for each extra zone. These are linear sequences - patterns where the same amount gets added each time.

Look at this sequence: 3, 7, 11, 15, 19... Notice how we add 4 each time. That's the key feature of linear sequences - the gap between consecutive terms stays constant.

The problem is obvious: what if you need the 50th term? You don't want to write out 50 numbers. That's where the nth term formula becomes your shortcut. The nth term formula allows you to find any term in the sequence without having to list all the previous numbers, streamlining the process significantly. Understanding this formula is essential for efficiently handling linear sequences.

2. The Formula Toolkit

Every linear sequence follows the same pattern. Once you spot the first term and the common difference, you can build a formula that works for any position.

The concept becomes clearer when you realise that the nth term equals the first term plus the product of (n-1) and the common difference. The number of steps is always one less than the term position.

Many students prefer the alternative form: $a_n = dn + (a - d)$. This shows the sequence as a straight line equation where the coefficient of n is the common difference.

3. Step-by-Step Method

Finding the nth term follows a reliable three-step process. You identify the pattern, build the formula, then test it works.

Let's work through the sequence 5, 8, 11, 14, 17... This looks linear, but we need to prove it and find the formula.

Worked example: Finding the nth Term Formula

Find the nth term of the sequence: 5, 8, 11, 14, 17...

1. $8 - 5 = 3$ $11 - 8 = 3$ $14 - 11 = 3$ $17 - 14 = 3$ — All differences are 3, so this is linear with d = 3
2. $a = 5$ (first term) $d = 3$ (common difference) — These are the building blocks of our formula
3. $a_n = a + (n-1)d$ $a_n = 5 + (n-1) \times 3$ — Replace a with 5 and d with 3
4. $a_n = 5 + 3n - 3$ $a_n = 3n + 2$ — Expand the brackets and collect like terms
5. When $n = 1$: $3(1) + 2 = 5$ ✓ When $n = 2$: $3(2) + 2 = 8$ ✓ — Always check your formula gives the correct first few terms

Answer: $a_n = 3n + 2$

Notice how the formula $a_n = 3n + 2$ tells you everything about the sequence. The coefficient of n (which is 3) is the common difference. The constant term (2) is what you get when you substitute n = 0.

4. Practice Problems

In this lesson, we will explore concepts related to linear sequences and their nth terms, including practice problems and common pitfalls. Let's begin!

The third problem shows a decreasing sequence. When the common difference is negative, your sequence goes down instead of up. The method stays exactly the same.

Worked example: Working Backwards from the nth Term

A sequence has nth term $a_n = 5n - 2$. Is 73 a term in this sequence?

1. $5n - 2 = 73$ — If 73 is in the sequence, then 5n - 2 must equal 73 for some value of n
2. $5n = 73 + 2$ $5n = 75$ $n = 15$ — Add 2 to both sides, then divide by 5
3. $n = 15$ — Since 15 is a positive integer, 73 is the 15th term
4. $a_{15} = 5(15) - 2 = 75 - 2 = 73$ ✓ — Always substitute back to check your working

Answer: Yes, 73 is the 15th term of the sequence

Let's practice identifying the nth term of linear sequences with the following fill-in-the-blank exercise.

5. Exam Techniques and Common Traps

Examiners love to test whether you truly understand sequences or just memorise formulas. They'll throw curveballs like negative differences, fractional terms, or asking you to work backwards.

The biggest trap is arithmetic errors when expanding $(n-1)d$. Students often forget to distribute the negative sign or make mistakes with negative common differences.

Another common error is confusing the position number with the term value. Remember: n is the position (1st, 2nd, 3rd...) and $a_n$ is the actual value at that position.

Tricky Exam Question

The graph shows how linear sequences relate to straight line graphs. The sequence 5, 8, 11, 14, 17... with nth term $3n + 2$ appears as points on the line $y = 3x + 2$. The gradient (3) is the common difference.

Now, let's work on some numerical examples to reinforce our understanding of linear sequences. Solve the following: 1. What is the 10th term of the sequence defined by $a_n = 2n + 1$?