Sequences and Nth Term Shortcuts

Sequences in IGCSE Maths

Sequences are ordered lists of numbers that follow a rule. Finding the nth term — a formula that generates any term in the sequence from its position number — is one of the most frequently tested skills in IGCSE Maths. It appears on both the Core and Extended papers.

Linear Sequences (Constant Difference)

A linear sequence has a constant difference between consecutive terms. For example: 3, 7, 11, 15, 19… has a common difference of 4.

The nth term of a linear sequence has the form: nth term = dn + c, where d is the common difference and c is a constant you need to find.

Shortcut method:

Find the common difference, d. In our example, d = 4.
Write the formula as 4n + c.
Use the first term to find c. When n = 1, the term is 3: 4(1) + c = 3, so c = −1.
The nth term is 4n − 1.

Verification: 4(1) − 1 = 3, 4(2) − 1 = 7, 4(3) − 1 = 11. Correct.

This method works for any linear sequence, including those with negative or fractional common differences.

Recognising Non-Linear Sequences

If the differences between terms are not constant, the sequence is not linear. Calculate the second differences (differences of the differences). If the second differences are constant, the sequence is quadratic.

Example: 2, 6, 14, 26, 42…

First differences: 4, 8, 12, 16
Second differences: 4, 4, 4 — constant

This is a quadratic sequence.

Quadratic Sequences

A quadratic sequence has the nth term in the form: an² + bn + c.

To find a, b, and c:

The second difference = 2a. If the second difference is 4, then a = 2.
Substitute n = 1, 2, 3 into an² to create a “comparison sequence”: 2, 8, 18, 32, 50…
Subtract this from the original sequence to get a new sequence: 0, −2, −4, −6, −8…
This new sequence is linear. Find its nth term: −2n + 2.
Combine: nth term = 2n² − 2n + 2.

Verification: 2(1) − 2 + 2 = 2, 2(4) − 4 + 2 = 6, 2(9) − 6 + 2 = 14. Correct.

Geometric Sequences

A geometric sequence has a constant ratio between consecutive terms. For example: 3, 6, 12, 24, 48… has a common ratio of 2.

The nth term is: a × rⁿ⁻¹, where a is the first term and r is the common ratio.

For our example: nth term = 3 × 2ⁿ⁻¹

Check: 3 × 2⁰ = 3, 3 × 2¹ = 6, 3 × 2² = 12. Correct.

Geometric sequences are tested on the Extended paper and often appear in contexts like population growth, depreciation, and compound interest.

Special Sequences to Recognise

Some sequences appear frequently and are worth memorising:

Square numbers: 1, 4, 9, 16, 25… → nth term = n²
Cube numbers: 1, 8, 27, 64, 125… → nth term = n³
Triangular numbers: 1, 3, 6, 10, 15… → nth term = n(n+1)/2
Fibonacci-type: each term is the sum of the two previous terms (you cannot write a simple nth term for this)
Powers of 2: 2, 4, 8, 16, 32… → nth term = 2ⁿ

Finding Specific Terms

Once you have the nth term formula, you can find any term by substituting the position number.

If the nth term is 3n² − 5n + 1, find the 20th term: 3(400) − 5(20) + 1 = 1200 − 100 + 1 = 1101

Is a Number in the Sequence?

A common exam question asks whether a particular number is in the sequence. Set the nth term equal to that number and solve for n. If n is a positive integer, the number is in the sequence.

Example: Is 150 a term in the sequence with nth term 4n − 2?

4n − 2 = 150 4n = 152 n = 38

Since 38 is a positive integer, yes, 150 is the 38th term of the sequence.

Example: Is 100 a term in the sequence 3, 7, 11, 15…?

nth term = 4n − 1. Set 4n − 1 = 100, so 4n = 101, n = 25.25. Since this is not an integer, 100 is not in the sequence.

Sum of Terms

On the Extended paper, you might be asked to find the sum of the first n terms of a series. For an arithmetic series (linear sequence):

Sum = (n/2)(first term + last term) or Sum = (n/2)(2a + (n−1)d)

For a geometric series:

Sum = a(rⁿ − 1)/(r − 1) when r > 1

These formulas may be given on the paper or may need to be recalled depending on the specific question.

Pattern Recognition Tips

When you see a sequence in the exam:

Calculate the first differences
If constant — it is linear (use the dn + c method)
If not constant, calculate second differences
If second differences are constant — it is quadratic
If neither, check for a constant ratio (geometric)
If none of these, look for patterns involving squares, cubes, or other known sequences

Common Mistakes

Using the wrong value of n (the first term corresponds to n = 1, not n = 0, unless stated otherwise)
Arithmetic errors when calculating differences
Forgetting to check your formula by substituting back
Confusing the common difference with the first term
Not simplifying the final formula
For quadratic sequences, forgetting that the coefficient of n² is half the second difference, not the second difference itself

Practice Plan

Sequences are a quick topic to practise. Each evening, try:

Two linear nth term questions (2 minutes)
One quadratic nth term question (5 minutes)
One “is this number in the sequence?” question (2 minutes)
One geometric sequence question (3 minutes)

Regular practice builds the pattern recognition that makes these questions feel automatic on exam day.

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