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Revision guide · IGCSE

IGCSE Maths sequences: nth term of linear, quadratic and geometric sequences

The nth term is a formula that gives any term of a sequence from its position n. For a linear sequence, the nth term is dn + c, where d is the common difference. For a quadratic sequence, the second differences are constant and equal to 2a, which gives the an^2 part. Cambridge 0580 Core expects linear, simple quadratic and simple cubic nth terms (C2.7); Extended adds exponential sequences and combinations (E2.7). Edexcel 4MA1 Foundation covers term-to-term rules and linear nth terms (3.1); Higher adds arithmetic sequences written as a + (n - 1)d and the sum of an arithmetic series, but does not list quadratic nth terms.

Facts checked:

At a glance

Cambridge 0580
Core C2.7; Extended E2.7
Edexcel 4MA1
Foundation 3.1A to C; Higher 3.1A to C
Linear rule
nth term = dn + (first term - d)
Series formula
Sn = n/2[2a + (n - 1)d], Edexcel Higher sheet

What each tier expects

Board and tierReferenceWhat can be asked
Cambridge 0580 CoreC2.7Continue sequences, term-to-term rules, nth term of linear, simple quadratic (e.g. 2, 5, 10, 17) and simple cubic sequences
Cambridge 0580 ExtendedE2.7Linear, quadratic, cubic and exponential sequences and simple combinations; subscript notation such as Tn
Edexcel 4MA1 Foundation3.1A to 3.1CTerm-to-term and position-to-term rules; nth term of arithmetic sequences such as 2n - 1
Edexcel 4MA1 Higher3.1A to 3.1CFirst term a and common difference d, nth term a + (n - 1)d, sum of the first n terms of an arithmetic series

The key ideas

Start every sequence question by writing the differences between consecutive terms underneath. If the first differences are all the same, the sequence is linear. If the first differences change but the second differences are constant, it is quadratic. If the terms multiply by the same number each time, it is geometric (exponential), and Cambridge Extended can ask for a formula such as 3 × 2^(n - 1).

For a linear sequence with common difference d, the nth term is dn + c. Find c by comparing with the first term: c = first term - d. Edexcel Higher writes the same idea as a + (n - 1)d, where a is the first term; expand it and you get dn + (a - d), identical.

For a quadratic sequence, half the second difference gives the coefficient of n^2. Subtract that an^2 from each term, and what is left is a linear sequence whose nth term you already know how to find.

To test whether a number is in a sequence, set the nth term equal to it and solve. If n comes out as a positive whole number, it is a term; otherwise it is not.

Worked example 1: nth term of 5, 9, 13, 17, ... and is 150 a term?

  1. Differences are all 4, so the sequence is linear with d = 4. The nth term is 4n + c.
  2. When n = 1 the term is 5, so 4 + c = 5 and c = 1. The nth term is 4n + 1.
  3. Check n = 4: 16 + 1 = 17. Correct.
  4. Is 150 a term? Solve 4n + 1 = 150: 4n = 149, n = 37.25. This is not a whole number, so 150 is not in the sequence. (101 is: 4n + 1 = 101 gives n = 25.)

Worked example 2: nth term of the quadratic sequence 3, 10, 21, 36, 55

  1. First differences: 7, 11, 15, 19. Second differences: 4, 4, 4. Constant, so the sequence is quadratic.
  2. Half the second difference is 2, so the nth term starts 2n^2.
  3. Subtract 2n^2 (which is 2, 8, 18, 32, 50) from each term: 1, 2, 3, 4, 5.
  4. The remainder is the linear sequence n. So the nth term is 2n^2 + n.
  5. Check n = 3: 2(9) + 3 = 21. Correct.

Worked example 3 (Edexcel Higher): arithmetic series

  1. Find the sum of the first 50 terms of 4 + 7 + 10 + 13 + ...
  2. First term a = 4, common difference d = 3, n = 50.
  3. Sn = n/2[2a + (n - 1)d] = 25[8 + 49 × 3] = 25[8 + 147] = 25 × 155 = 3875.
  4. Check another way: the 50th term is 4 + 49 × 3 = 151, and the sum is 50 × (4 + 151)/2 = 50 × 77.5 = 3875. Correct.
  5. A related Higher question: the 2nd term is 7 and the 5th term is 19. From the 2nd to the 5th term is 3 differences, so 3d = 12, d = 4 and a = 3.

Common mistakes that cost marks

  • Giving the term-to-term rule ('add 4') when the question asks for the nth term.
  • Writing n + 4 instead of 4n + 1 for a sequence that goes up in 4s.
  • For quadratic sequences, using the full second difference as the n^2 coefficient instead of half of it.
  • Forgetting that a decreasing linear sequence has a negative d: 7, 4, 1, -2 has nth term -3n + 10.
  • Concluding a number is a term when n comes out as a decimal or negative.
  • Mixing up a + (n - 1)d with a + nd in arithmetic sequences.

Exam technique and how a tutor helps

Sequences questions are usually early in the paper and quick, so they are good marks to bank. Always write out the differences, because that line earns credit if the formula goes wrong, and always test your nth term on two given terms before moving on. On non-calculator papers the arithmetic is designed to be manageable, so if you get awkward numbers, recheck.

Cambridge Extended can combine types, for example a sequence whose terms are n^2 + 2^n. Spot these by comparing the given terms with familiar sequences: squares, cubes, powers of 2, triangular numbers.

In lessons, a tutor will often have a student explain why the nth term of a linear sequence has the difference in front of n, rather than memorise it. Students who understand that link find the quadratic method and the arithmetic series formula much easier, and a one-to-one lesson leaves room to build that understanding before drilling past-paper questions.

Self-check: can you do these?

  • Find the nth term of 7, 4, 1, -2. (Answer: 10 - 3n)
  • Find the nth term of 6, 15, 28, 45. (Answer: 2n^2 + 3n + 1)
  • Find the nth term of 2, 9, 28, 65. (Answer: n^3 + 1)
  • Find a formula for the nth term of 3, 6, 12, 24. (Answer: 3 × 2^(n - 1))
  • The nth term is 4n + 3. Write down the first three terms. (Answer: 7, 11, 15)

Common questions

What is the difference between a term-to-term rule and an nth term?

A term-to-term rule tells you how to get from one term to the next, such as 'add 4'. The nth term gives any term directly from its position, such as 4n + 1, so you can find the 100th term without listing the first 99.

Are quadratic sequences on the Edexcel International GCSE?

The Edexcel 4MA1 specification lists linear (arithmetic) sequences and arithmetic series, not quadratic nth terms. Cambridge 0580 includes simple quadratic sequences even at Core. If your child sits Edexcel, quadratic sequences are still useful practice but not a listed requirement.

Do I need to memorise the arithmetic series formula?

For Edexcel Higher it appears on the formulae sheet. You still need to know what a, d and n stand for and use it correctly.

How do I recognise an exponential sequence?

Divide each term by the one before. If the ratio is constant, such as 2 or 0.5, the sequence is geometric and its nth term has the form a × r^(n - 1). This is Cambridge Extended content.

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Sources

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