Sequences and proof at GCSE
A sequence is an ordered list of numbers that follow a rule. At GCSE you need to work with two main types:
- Arithmetic sequences — a constant amount is added or subtracted each time (called the common difference, d)
- Geometric sequences — each term is multiplied by a constant (called the common ratio, r)
You also need to find the nth term — a formula that generates any term in the sequence — and recognise special sequences like square numbers, cube numbers, triangular numbers and Fibonacci-type sequences.
Algebraic proof appears at Higher tier. You use algebra to prove a statement is always true, rather than just showing it works for a few examples.
Arithmetic sequences and the nth term
In an arithmetic sequence the difference between consecutive terms is constant. The nth term formula is:
nth term = a + (n − 1)d
where d is the common difference and a is the first term. Equivalently: start with the common difference as the coefficient of n, then adjust the constant so the formula gives the correct first term.
Worked Example 1 — Finding the nth term of an arithmetic sequence
Find the nth term of the sequence: 7, 11, 15, 19, 23, …
1
Find the common difference: 11 − 7 = 4, so d = 4
2
Substitute into nth term = a + (n − 1)d: nth term = 7 + (n − 1) × 4
3
Expand and simplify: 7 + 4n − 4 = 4n + 3
4
Check: when n = 2: 4(2) + 3 = 11 ✓ when n = 5: 4(5) + 3 = 23 ✓
Answernth term = 4n + 3
Worked Example 2 — Using the nth term
The nth term of a sequence is 5n − 2. (a) Find the 10th term. (b) Is 73 a term in the sequence?
1
(a) Substitute n = 10: 5(10) − 2 = 50 − 2 = 48
2
(b) Set 5n − 2 = 73 and solve: 5n = 75 → n = 15
3
Since n = 15 is a positive integer, 73 is the 15th term in the sequence
Answer(a) 48 (b) Yes — 73 is the 15th term
Geometric sequences
In a geometric sequence each term is multiplied by the same constant — the common ratio r. To find r, divide any term by the one before it. Geometric sequences can grow very rapidly (r > 1), decay towards zero (0 < r < 1), or alternate in sign (r < 0).
Worked Example 3 — Identifying and extending a geometric sequence
For the sequence 3, 6, 12, 24, …: (a) Find the common ratio. (b) Find the next two terms. (c) Find the 8th term.
1
(a) Common ratio: 6 ÷ 3 = 2. Check: 12 ÷ 6 = 2 ✓
2
(b) Multiply the last known term by 2 repeatedly: 24 × 2 = 48, then 48 × 2 = 96
3
(c) The nth term of a geometric sequence is ar^(n−1). Here a = 3, r = 2: 8th term = 3 × 2⁷ = 3 × 128 = 384
Answer(a) r = 2 (b) 48, 96 (c) 384
Algebraic proof
In algebraic proof you use algebra — not just numerical examples — to show that a statement is always true. The key is to express integers in a general form:
- Any integer: n
- Any even number: 2n
- Any odd number: 2n + 1
- Consecutive integers: n, n + 1, n + 2
- Consecutive even numbers: 2n, 2n + 2, 2n + 4
Show the result algebraically, then write a clear conclusion explaining what you have proved.
Worked Example 4 — Algebraic proof
Prove that the sum of any two consecutive odd numbers is always divisible by 4
1
Let the first odd number be 2n + 1. The next consecutive odd number is 2n + 3
2
Find their sum: (2n + 1) + (2n + 3) = 4n + 4
3
Factorise: 4n + 4 = 4(n + 1)
4
Since 4(n + 1) is 4 multiplied by an integer, it is always divisible by 4
Answer4(n + 1) is always a multiple of 4, so the sum of any two consecutive odd numbers is always divisible by 4
Practice questions
Work through each question before checking the answers. Difficulty is shown for each question.
Foundation (Grade 3–5)
Q1Find the next three terms in the sequence: 5, 8, 11, 14, …Foundation
Q2Find the nth term of the arithmetic sequence: 3, 7, 11, 15, 19, …Foundation
Q3The nth term of a sequence is 6n − 1. Find the first five terms and the 20th term.Foundation
Q4Is 50 a term in the sequence with nth term 4n + 2? Show your working.Foundation
Q5Write down the first four terms of the geometric sequence with first term 5 and common ratio 3.Foundation
Higher (Grade 5–7)
Q6Find the nth term of: 2, −1, −4, −7, −10, …Higher
Q7A geometric sequence has first term 2 and common ratio 5. Find the 6th term.Higher
Q8The 3rd term of an arithmetic sequence is 17 and the 7th term is 37. Find the first term and common difference.Higher
Q9The nth term of a sequence is n² + 2. Find the first four terms and state whether the sequence is arithmetic, geometric or neither.Higher
Q10A geometric sequence has terms 4, 12, 36, … . Which term of the sequence first exceeds 3000?Higher
Higher — Hard (Grade 8–9)
Q11Prove that the product of two consecutive even numbers is always divisible by 8.Grade 8–9
Q12Prove that (n + 3)² − (n + 1)² is always a multiple of 4.Grade 8–9
Q13Prove that the sum of three consecutive integers is always a multiple of 3.Grade 8–9
Q14Show that n² + n is always even for any positive integer n.Grade 8–9
Q15Tom says: "The square of any odd number is always odd." Prove that Tom is correct.Grade 8–9
Answers
Foundation (Q1–Q5)
Q117, 20, 23(common difference = 3; add 3 each time)
Q2nth term = 4n − 1(a = 3, d = 4; nth term = 3 + (n−1)×4 = 3 + 4n − 4 = 4n − 1)
Q3First five terms: 5, 11, 17, 23, 29; 20th term = 119(6(20) − 1 = 119)
Q4Yes — 50 is the 12th term(4n + 2 = 50 → 4n = 48 → n = 12; since n is a positive integer, 50 is in the sequence)
Q55, 15, 45, 135(multiply by 3 each time)
Higher (Q6–Q10)
Q6nth term = −3n + 5(a = 2, d = −3; nth term = 2 + (n−1)×(−3) = 2 − 3n + 3 = −3n + 5)
Q76th term = 6250(ar^(n−1) = 2 × 5⁵ = 2 × 3125 = 6250)
Q8First term = 7, common difference = 5(7th − 3rd = 4d: 37 − 17 = 20 → d = 5; 3rd term = a + 2d: 17 = a + 10 → a = 7)
Q9Terms: 3, 6, 11, 18; neither arithmetic nor geometric(differences: 3, 5, 7 — not constant; ratios: 2, 11/6, 18/11 — not constant)
Q10The 7th term (= 4 × 3⁶ = 2916; 8th term = 8748 > 3000, so the 7th term is 2916 < 3000, the 8th term first exceeds 3000)(4×3⁷ = 8748 > 3000; 4×3⁶ = 2916 < 3000)
Higher — Hard (Q11–Q15)
Q11Let the two consecutive even numbers be 2n and 2n + 2. Product = 2n(2n + 2) = 4n(n + 1). Since one of n or n+1 is always even, n(n+1) is always even, so 4n(n+1) is divisible by 8.
Q12(n+3)² − (n+1)² = n²+6n+9 − (n²+2n+1) = 4n + 8 = 4(n+2). This is always a multiple of 4.
Q13Let the integers be n, n+1, n+2. Sum = 3n + 3 = 3(n+1). This is always a multiple of 3.
Q14n² + n = n(n+1). One of n or n+1 is always even, so their product is always even.
Q15Let the odd number be 2n+1. (2n+1)² = 4n²+4n+1 = 4n(n+1)+1. Since 4n(n+1) is even, adding 1 gives an odd number. So the square of any odd number is always odd.
Common mistakes
These are the errors Alamin sees most frequently with sequences and proof at GCSE. Recognising them now will save you marks in the exam.
Common Mistake 1
Getting the constant wrong in the nth term
Students find d correctly but then write, for example, 3n instead of 3n + 1 without checking. Always substitute n = 1 into your nth term and check it gives the first term. If it doesn't, adjust the constant.
Common Mistake 2
Confusing arithmetic and geometric sequences
In an arithmetic sequence you add the same number each time. In a geometric sequence you multiply by the same number. A sequence like 2, 6, 18, 54 is geometric (×3), not arithmetic (+16 — that's clearly wrong). Always check by testing both add and multiply.
Common Mistake 3
Checking examples instead of proving algebraically
In proof questions, writing "when n = 1 it works, when n = 2 it works…" scores zero marks. You must use algebra with general expressions like 2n, 2n+1 or n, n+1, n+2 to prove the result works for ALL integers.
Common Mistake 4
Using the wrong expressions for odd and even numbers
An even number is 2n, not 2n + 2 (which is the next even number). An odd number is 2n + 1 (or 2n − 1). Using n for "any even number" is wrong — n could be any integer, including odd ones.
Common Mistake 5
Forgetting to write a conclusion in proof questions
After the algebra, you must write a sentence explaining what you have shown. For example: "Since 4(n+1) is always a multiple of 4, the statement is proved." Without this conclusion, you will typically lose the final mark even if the algebra is perfect.
Common Mistake 6
Applying the arithmetic nth term formula to geometric sequences
The formula nth term = a + (n − 1)d only works for arithmetic sequences. For geometric sequences the nth term is ar^(n−1). Using the arithmetic formula on a geometric sequence will give completely wrong answers.
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