Two kinds of rule, and the sequences to recognise
A term-to-term rule says how to get from one term to the next: "start at 4 and add 3 each time". It is easy to use but slow — finding the 100th term means working through the previous 99.
A position-to-term rule, the nth term, gives any term directly from its position. With nth term = 3n + 1, the 100th term is 3(100) + 1 = 301 in one step. That is why exam questions ask for it.
The diagnostic is quick. Work out the differences between consecutive terms. If they are constant, the sequence is linear (arithmetic). If they are not, work out the differences of those differences — if that second difference is constant, it is quadratic. If neither, check whether each term divides into the next to give a constant ratio, which makes it geometric.
A geometric ratio can be a fraction, giving a decreasing sequence such as 80, 40, 20, 10, or a surd, as in 2, 2√2, 4.
- Term-to-term rule
- a rule saying how to get from one term of a sequence to the next
- Position-to-term rule
- the nth term: a rule giving any term of a sequence directly from its position
The nth term of a linear sequence
For a linear sequence the common difference is the coefficient of n. Then adjust with a constant to make the first term come out right.
For 4, 7, 10, 13: the common difference is 3, so the rule starts 3n. Testing n = 1 gives 3, but the first term is 4, so add 1. The nth term is 3n + 1.
A quick check on the second and third terms confirms it: 3(2) + 1 = 7 ✓ and 3(3) + 1 = 10 ✓.
A decreasing sequence has a negative difference, so 20, 17, 14, 11 has nth term −3n + 23. Write it as 23 − 3n if that reads more naturally.
The nth term also answers the standard follow-up: is a given number in the sequence? Set the rule equal to it and solve. If n comes out as a positive whole number, it is; if not, it is not — and saying which term it is, or why it cannot be, is what earns the mark.
The nth term of a quadratic sequenceHigher tier
A quadratic sequence has a constant second difference, and its nth term has the form an² + bn + c.
The method is reliable enough to follow mechanically:
1. Find the second difference and halve it — that gives a 2. Write out an² for each position, and subtract it from the original sequence 3. What remains is a linear sequence; find its nth term as bn + c 4. Add the two parts together
Worked example
Find the nth term of 3, 8, 17, 30, 47.
- First differences: 5, 9, 13, 17. Not constant, so it is not linear.
- Second differences: 4, 4, 4. Constant, so it is quadratic.
- Halve the second difference: 4 ÷ 2 = 2, so a = 2 and the sequence contains 2n².
- Write out 2n² for n = 1 to 5: 2, 8, 18, 32, 50.
- Subtract from the original: 3 − 2 = 1, 8 − 8 = 0, 17 − 18 = −1, 30 − 32 = −2, 47 − 50 = −3.
- That remainder — 1, 0, −1, −2, −3 — is linear with common difference −1, so it starts −n; testing n = 1 gives −1, but we need 1, so add 2. The remainder is −n + 2.
- Add the parts together.
Answer: nth term = 2n² − n + 2. Check with n = 4: 2(16) − 4 + 2 = 32 − 2 = 30 ✓
Common mistake
The coefficient of n² is half the second difference, not the second difference itself. Checking one term at the end catches it immediately.
- Quadratic sequence
- a sequence with a constant second difference, whose nth term has the form an² + bn + c
Before you practise
- A term-to-term rule gets you to the next term; a position-to-term rule (the nth term) gets you to any term directly.
- Constant first difference means linear; constant second difference means quadratic; constant ratio means geometric.
- Triangular numbers: 1, 3, 6, 10, 15, with nth term n(n + 1) ⁄ 2.
- A Fibonacci-type sequence adds the two previous terms, from any two starting values.
- A geometric ratio can be a fraction or a surd.
- For a linear sequence, the common difference is the coefficient of n; then adjust the constant.
- For a quadratic sequence, halve the second difference to get a, subtract an², and find the nth term of what is left.
- To test whether a number is in a sequence, set the nth term equal to it and check n is a positive whole number.
- Always verify your rule on one term you did not use to build it.