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To use a term-to-term rule, start with the first term and apply the rule repeatedly. Free GCSE AQA Maths notes with step-by-step methods and 13 FAQs.
Generating Sequence Terms is part of Sequences in GCSE AQA Higher Maths.
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GCSE AQA Higher
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To use a term-to-term rule, start with the first term and apply the rule repeatedly. Free GCSE AQA Maths notes with step-by-step methods and 13 FAQs.
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Use this lesson to stay inside Sequences while connecting the detail back to the wider Algebra topic.
Read the core explanations from Generating Sequence Terms before testing yourself from memory.
Sequences appear everywhere — from the pattern of seats in a theatre to the way compound interest grows in a bank account. At GCSE, understanding sequences means you can describe, predict, and analyse patterns using algebra rather than listing terms one by one.
There are two fundamentally different ways to describe a sequence. The first tells you how to move from one term to the next. The second tells you how to jump straight to any term you want. Both are tested at GCSE, and knowing which one you are dealing with is the first skill to master.
Think of it like directions. A term-to-term rule is like saying "turn left, then turn right, then go straight" — you follow each step in order. A position-to-term rule is like a postcode: you can find any address directly, without retracing the whole journey.
A term-to-term rule always needs a starting point. Without knowing where the sequence begins, the rule alone is not enough to pin down a unique sequence. For example, "add 3 each time" could describe 1, 4, 7, 10, ... or 5, 8, 11, 14, ... depending on the first term.
The rule can be additive (add or subtract a fixed amount), multiplicative (multiply or divide by a fixed amount), or more complex. At GCSE, you will mostly meet additive and multiplicative rules, but you should be comfortable recognising both. Let's apply these concepts in the next set of exercises.
Here is how to generate terms systematically. You are given: first term = 2, term-to-term rule = multiply by 3. You apply the rule step by step to build the sequence.
Worked example: Generating Terms Using a Term-to-Term Rule
The first term of a sequence is 2. The term-to-term rule is: multiply by 3. Find the first five terms.
Answer: 2, 6, 18, 54, 162
Notice that each term is exactly three times the one before it. This is a geometric sequence — a sequence where you multiply by the same number each time. Contrast this with an arithmetic sequence, where you add the same number each time.
Now try a subtraction-based rule. The first term is 50 and the term-to-term rule is: subtract 7. This gives a decreasing arithmetic sequence. The terms will eventually become negative — that is perfectly valid and examiners do test this.
Worked example: Generating a Decreasing Sequence
The first term of a sequence is 50. The term-to-term rule is: subtract 7. Find the first five terms.
Answer: 50, 43, 36, 29, 22
This is where students often slip up. Can you spot the key difference between these two types of term-to-term rules before moving on?
The position-to-term rule is the more powerful of the two approaches. With a term-to-term rule, finding the 100th term means applying the rule 99 times. With a position-to-term rule, you substitute $n = 100$ and you are done in one step.
The most common position-to-term rules at GCSE are linear expressions like $3n + 1$ or $5n - 2$. You might also meet quadratic rules like $n^2 + 3$ at higher tier. The method for using them is always the same: substitute the position number for $n$ and evaluate.
Here is the key insight: $a$ is the number you add each time (the common difference), and $b$ shifts the whole sequence up or down. Once you can read these two values from a rule, you can describe the sequence without listing a single term.
Worked example: Generating Terms from a Position-to-Term Rule
The nth term of a sequence is $4n - 1$. Find the first four terms and the 20th term.
Answer: First four terms: 3, 7, 11, 15. The 20th term is 79.
You will notice the sequence 3, 7, 11, 15 increases by 4 each time — matching the coefficient of $n$ in the rule $4n - 1$. This is always the case for linear sequences: the coefficient of $n$ equals the common difference.
Now let us try a rule with a negative coefficient, which produces a decreasing sequence. The method is identical — substitute and evaluate — but you must be careful with negative multiplication.
Worked example: Using a Position-to-Term Rule with a Negative Coefficient
The nth term of a sequence is $-3n + 20$. Find the first four terms and the 10th term.
Answer: First four terms: 17, 14, 11, 8. The 10th term is −10.
The sequence decreases by 3 each time, which matches the $-3$ coefficient. Negative coefficients always produce decreasing sequences — a quick sanity check you can use in the exam.
A linear position-to-term rule like $T_n = 4n - 1$ produces a straight-line pattern when you plot term value against position number. This is because the rule is a linear function of $n$. The gradient of that line equals the common difference, and the vertical intercept relates to the constant $b$.
Plotting the first few terms of $T_n = 4n - 1$ gives the points (1, 3), (2, 7), (3, 11), (4, 15). These lie on a straight line with gradient 4. This visual connection between sequences and linear graphs is a powerful way to check your working.
The graph above shows the line $y = 4x - 1$. The sequence terms sit exactly on this line at integer positions $x = 1, 2, 3, 4, ...$. Notice the line does not pass through the origin — the $-1$ shifts it downward. This is the visual meaning of the constant $b$ in $T_n = an + b$.
The only way to make these methods automatic is to practise them on varied examples. The problems below cover both rule types and include a mix of increasing and decreasing sequences. Try each one before checking the solution.
Pay attention to whether you are given a term-to-term rule or a position-to-term rule — the method is different for each. This distinction is exactly what examiners test.
Here is a passage to consolidate the key vocabulary before the tips block. Fill in the blanks to check you have the language right — examiners reward precise mathematical terminology.
Before the exam section, here is a compact summary of the key rules and formulas for generating sequence terms. Use this as a rapid retrieval tool when revising.
Generating sequence terms is a reliable source of marks in GCSE Higher papers — but only if you avoid the traps that catch students out every year. The most common errors are not conceptual; they are procedural. You know the method, but a small slip costs you the mark.
The single most frequent mistake is confusing the two types of rule. If you are given a position-to-term rule and you treat it like a term-to-term rule (adding the coefficient each time instead of substituting), you will generate a completely wrong sequence. Always identify which type of rule you have before you start.
In questions without a calculator, you must show all arithmetic working. Write out each substitution explicitly: $T_5 = 4(5) - 1 = 20 - 1 = 19$. Do not skip steps — method marks are available even if your final answer is wrong. Next, let’s consider how different rules affect the sequence.
On calculator-allowed exam questions , you can use your calculator to evaluate the expression after substituting, but you must still write the substitution step to earn the method mark.
Here is a cloze passage to consolidate the key vocabulary before the tips block. Fill in the blanks to check you have the language right — examiners reward precise mathematical terminology.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Start with the first term and apply the rule repeatedly. With a first term of 2 and the rule multiply by 3, the sequence is 2, 6, 18, 54, 162; with a first term of 50 and the rule subtract 7, it is 50, 43, 36, 29, 22.
A geometric sequence multiplies by the same number each time, such as 2, 6, 18, 54, 162, while an arithmetic sequence adds the same number each time, such as 3, 7, 11, 15. Arithmetic sequences can also decrease, such as 50, 43, 36, 29.
A position-to-term rule, or nth term, is an expression in n that gives the value of the term at position n directly. Substituting n = 1 gives the first term and n = 100 gives the 100th term, without needing any previous terms.
Substitute n = 20 into the position-to-term rule. For the nth term 4n - 1, the 20th term is 4(20) - 1 = 80 - 1 = 79, with no need to list the first 19 terms.
A term-to-term rule gives each term from the previous one, so it needs a starting term. A position-to-term rule, or nth term, gives any term directly: substitute n = 1 for the first term, n = 2 for the second, and so on.
For a linear sequence, the coefficient of n equals the common difference. So 4n - 1 gives 3, 7, 11, 15, increasing by 4 each time, and -3n + 20 gives 17, 14, 11, 8, decreasing by 3 each time.
The first five terms are 2, 6, 18, 54 and 162. Each term is the previous term multiplied by 3, so 2 × 3 = 6, 6 × 3 = 18, 18 × 3 = 54 and 54 × 3 = 162, making it a geometric sequence.
The first four terms of 4n - 1 are 3, 7, 11 and 15, found by substituting n = 1, 2, 3 and 4. Substituting n = 20 gives the 20th term, 4(20) - 1 = 79.
The sequence -3n + 20 starts 17, 14, 11, 8 and decreases by 3 each time, matching the coefficient -3. The 10th term is -3(10) + 20 = -30 + 20 = -10, so the sequence has gone negative by position 10.
A term-to-term rule needs a first term because the rule alone does not pin down a unique sequence. 'Add 3 each time' could describe 1, 4, 7, 10, ... or 5, 8, 11, 14, ... depending on where the sequence begins.
A linear sequence such as 4n - 1 plots as points on a straight line when term value is plotted against position. The points (1, 3), (2, 7), (3, 11) and (4, 15) lie on y = 4x - 1, whose gradient 4 is the common difference.
The most common mistake is confusing the two types of rule, for example treating a position-to-term rule like a term-to-term rule and adding the coefficient each time instead of substituting. So always check which type of rule is given before starting.
Terms of a sequence can be negative, and decreasing sequences often become negative. Starting at 50 and subtracting 7 gives 50, 43, 36, 29, 22, which will eventually become negative.