Sequences and the nth Term

A sequence is an ordered list of numbers. Almost every question comes down to one skill: finding the nth term, a formula that generates any term you want without listing them all.

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1. Two different rules — don’t confuse them

The term-to-term rule tells you how to get from one term to the next. The nth term (position-to-term rule) gives you any term directly from its position.

For 5, 8, 11, 14, …

• term-to-term: “add 3”
• nth term: 3n + 2

These answer different questions. “Add 3” cannot tell you the 100th term without 97 more additions. The nth term gives it immediately: 3(100) + 2 = 302. Confusing the two was a repeated error, and questions usually specify which they want.

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2. Linear (arithmetic) sequences

A sequence with a constant difference between terms.

nth term = dn + c, where d is the common difference.

Method:

1. Find the common difference d
2. Write dn
3. Work out the constant: substitute n = 1 and adjust to hit the first term

Example: 5, 8, 11, 14, …

1. d = 3
2. Start with 3n → gives 3, 6, 9, 12
3. Each is 2 less than the sequence, so add 2
4. nth term = 3n + 2

Example: 13, 9, 5, 1, … (a decreasing sequence)

1. d = −4
2. −4n → gives −4, −8, −12
3. The sequence is 17 more each time
4. nth term = −4n + 17, usually written 17 − 4n

A decreasing sequence has a negative coefficient of n. Writing “+4” or “subtract 4” as the nth term instead of −4n + 17 was a documented error. The difference must carry its sign.

The nth term is an expression, not an instruction. “Plus 7” is not an nth term; 7n − 5 is.

Using the nth term:

• Find the 100th term: substitute n = 100
• Is 77 in the sequence? Set the nth term equal to 77 and solve. If n is a positive whole number, yes; if it’s a fraction or negative, no.

Example: is 77 in 3n + 2? 3n + 2 = 77 → 3n = 75 → n = 25. Yes, it is the 25th term. Is 100? 3n + 2 = 100 → n = 32.67 — not a whole number, so no.

n must be a positive integer. It is a position in a list, so answers like n = 32.67 or n = −4 mean the number is not in the sequence — and saying so is the answer.

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3. Quadratic sequences

The differences are not constant, but the second differences are.

Example: 3, 8, 15, 24, 35, …

• first differences: 5, 7, 9, 11
• second differences: 2, 2, 2 → constant, so it’s quadratic

Divide the second difference by 2 to get the coefficient of n².

Here 2 ÷ 2 = 1, so the sequence starts with n².

Method:

1. Find first and second differences
2. a = second difference ÷ 2 → gives the an² term
3. Subtract an² from each original term
4. What remains is a linear sequence — find its nth term as usual
5. Add the two parts together

Continuing the example:

n	1	2	3	4	5
term	3	8	15	24	35
n²	1	4	9	16	25
term − n²	2	4	6	8	10

The remainder 2, 4, 6, 8, 10 is linear with nth term 2n.

nth term = n² + 2n

Check with n = 3: 9 + 6 = 15 ✓

Always check your formula against two or three terms. It takes seconds and catches nearly every error in this method.

Halve the second difference — don’t use it directly. Using 2 as the coefficient instead of 1 was the single most common quadratic-sequence error.

A quadratic sequence is not a quadratic equation. You are finding a formula, not solving for x.

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4. Geometric sequences

Each term is found by multiplying by a constant — the common ratio r.

Example: 3, 6, 12, 24, … → r = 2

nth term = ar^(n−1), where a is the first term.

Here: 3 × 2^(n−1). Check n = 4: 3 × 2³ = 24 ✓

The power is n − 1, not n, because the first term has been multiplied zero times.

Find the ratio by dividing consecutive terms: r = second ÷ first. For 3, 6, 12: r = 6/3 = 2.

Ratios can be fractional (48, 24, 12 → r = ½) or negative (2, −6, 18 → r = −3, giving alternating signs).

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5. Other patterns worth recognising

Sequence	Pattern	nth term
1, 4, 9, 16, 25	square numbers	n²
1, 8, 27, 64	cube numbers	n³
2, 4, 6, 8	even	2n
1, 3, 5, 7	odd	2n − 1
1, 3, 6, 10, 15	triangular	n(n+1)/2
2, 4, 8, 16	powers of 2	2ⁿ

Cubic sequences (constant third difference) appear rarely; the method extends the same way, with the third difference ÷ 6 giving the n³ coefficient.

Pattern questions — matchsticks, tiles, growing shapes — are linear sequences in disguise. Count the objects in the first few diagrams, write them as a sequence, and find the nth term as normal.

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6. Scope note

Some lessons covered the sum of an arithmetic series, n/2 × (2a + (n−1)d).

This is not on the 0580 syllabus. Summing series belongs to Additional Mathematics (0606) and A-level. On 0580 you are asked to continue sequences and find nth terms, not to sum them. If you are sitting 0580 only, you can skip this formula — check your own syllabus document before spending time on it.

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7. Mistakes that cost marks

Giving the term-to-term rule when the nth term was asked for.

Writing an instruction (“add 3”) instead of an expression.

Losing the negative sign on a decreasing sequence.

Using the second difference directly instead of halving it.

Sign errors when calculating differences.

Not checking the formula against the original terms.

Forgetting n − 1 in a geometric nth term.

Concluding a number is in a sequence when n came out fractional or negative.

Substituting the term value for n, or vice versa.

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Frequently asked questions

What is the nth term? A formula giving any term of a sequence directly from its position n.

How do you find the nth term of a linear sequence? The coefficient of n is the common difference; then adjust the constant so n = 1 gives the first term.

How do you find the nth term of a quadratic sequence? Halve the second difference for the n² coefficient, subtract that part from each term, then find the nth term of the linear remainder.

How do I know if a sequence is quadratic? The second differences are constant.

What is the nth term of a geometric sequence? ar^(n−1), where a is the first term and r the common ratio.

How do I find the common ratio? Divide any term by the one before it.

How do I check whether a number is in a sequence? Set the nth term equal to it and solve for n. It is in the sequence only if n is a positive whole number.

What’s the difference between term-to-term and nth term? Term-to-term gets you from one term to the next; the nth term gets you any term directly.

Do I need the sum of an arithmetic series? Not for 0580 — that is 0606 material.

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Quick revision checklist

• I can tell a term-to-term rule from an nth term
• I give the nth term as an expression
• I can find the nth term of a linear sequence, including decreasing ones
• I keep the sign of the common difference
• I can use the nth term to find a specific term
• I can test whether a number belongs to a sequence, and interpret a non-integer n
• I can spot a quadratic sequence from constant second differences
• I halve the second difference for the n² coefficient
• I can complete the method by finding the linear remainder
• I check my formula against two or three terms
• I can find the common ratio and nth term of a geometric sequence
• I remember the power is n − 1
• I recognise squares, cubes, triangular numbers and powers of 2
• I can turn a diagram pattern into a sequence
• I know the sum formula is not on 0580

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These notes cover sequences in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.