Sequences: nth Term, Quadratic & Geometric

Sequences: nth Term, Quadratic & Geometric

A sequence is an ordered list of numbers that follows a rule. GCSE Maths covers arithmetic (linear), quadratic and geometric sequences.

Arithmetic (Linear) Sequences

An arithmetic sequence has a common difference (d) between consecutive terms.

Example: 5, 8, 11, 14, 17, ... (common difference = 3)

nth term formula: nth term = dn + (a − d)

Where d = common difference and a = first term. Equivalently: a + (n − 1)d.

Worked Example: Find the nth term of 7, 11, 15, 19, ...

• Common difference d = 4
• First term a = 7
• nth term = 4n + (7 − 4) = 4n + 3
• Check: 1st term = 4(1) + 3 = 7 ✓, 2nd term = 4(2) + 3 = 11 ✓

Worked Example: Is 150 a term in the sequence 3, 7, 11, 15, ...?

• nth term = 4n − 1
• Set 4n − 1 = 150 → 4n = 151 → n = 37.75
• Since n is not a whole number, 150 is not in the sequence

Quadratic Sequences (Higher)

A quadratic sequence has a second difference that is constant.

Example: 3, 9, 19, 33, 51, ...

Term	3	9	19	33	51
1st diff	6	10	14	18
2nd diff	4	4	4

Method to find the nth term:

1. The second difference = 2a, so a = second difference ÷ 2

2. Write out an² for each term and subtract from the original sequence

3. Find the nth term of the remaining linear sequence

4. Combine: nth term = an² + bn + c

Worked Example: Find the nth term of 3, 9, 19, 33, 51.

• Second difference = 4, so a = 2 → the n² coefficient is 2
• Subtract 2n² from each term:

n	1	2	3	4	5
Term	3	9	19	33	51
2n²	2	8	18	32	50
Remainder	1	1	1	1	1

• The remainder is constant (1), so the nth term = 2n² + 1

Geometric Sequences (Higher)

A geometric sequence has a common ratio (r) between consecutive terms — each term is multiplied by the same number.

Example: 2, 6, 18, 54, 162, ... (common ratio = 3)

nth term formula: nth term = a × r^(n−1)

Where a = first term, r = common ratio.

Worked Example: Find the 8th term of 5, 10, 20, 40, ...

• a = 5, r = 10 ÷ 5 = 2
• 8th term = 5 × 2⁷ = 5 × 128 = 640

Worked Example: A geometric sequence starts 256, 192, 144, ... Find the common ratio and the 5th term.

• r = 192 ÷ 256 = 0.75
• 5th term = 256 × 0.75⁴ = 256 × 0.31640625 = 81

Other Sequences to Recognise

• Triangular numbers: 1, 3, 6, 10, 15, ... (nth term = n(n+1)/2)
• Square numbers: 1, 4, 9, 16, 25, ... (nth term = n²)
• Cube numbers: 1, 8, 27, 64, 125, ... (nth term = n³)
• Fibonacci-type: each term is the sum of the two before it (1, 1, 2, 3, 5, 8, ...)

Exam Tips

• For arithmetic sequences, always find the common difference first
• For quadratic sequences, you MUST look at second differences
• The nth term must work for ALL terms — always check by substituting n = 1, 2, 3
• Geometric sequences can decrease (when 0 < r < 1) or alternate in sign (when r is negative)