Sequences.
Look beyond the next term: describe how the position controls the value.
Use term-to-term and nth-term rules · recognise linear, quadratic, cubic and exponential patterns
Position is different from the previous term
A sequence is an ordered list whose position matters. The first term has position 1, the second has position 2, and so on. A term-to-term rule tells you how to move along the list.
A position-to-term rule, often called the nth-term rule, calculates a term directly from its position. These are different jobs. Start by making a two-row table of positions and values, then look for how changes in position produce changes in value.
A term-to-term rule describes how to get the next term. An nth-term rule calculates a term directly from its position n, usually starting at n = 1.
Tn = 3n + 2. First differences are constant: +3. The nth term is 3n + 2.
Find the nth term of 5, 8, 11, 14, …
The difference is 3, so begin with 3n. At n = 1, 3n gives 3; add 2 to reach 5.
The 20th term is 3 × 20 + 2 = 62. To test membership, solve 3n + 2 = 80: n = 26, so 80 is a term.
Pause and explainWhat is the difference between “add 3” and “3n + 1”?
“Add 3” moves from a known term to the next. “3n + 1” calculates the term directly from position n.
Quadratics and cubics: take differences
If the first differences are not constant, compare the differences themselves. Constant second differences suggest a quadratic rule; constant third differences suggest a cubic rule in the intended polynomial pattern.
The differences help choose a possible family, but the proposed formula must still reproduce the listed terms. Build the main square or cube part, compare it with the original sequence, then account for the leftover pattern. Check several positions rather than just the first.
Find a rule for 2, 6, 12, 20, 30, …
First differences: 4, 6, 8, 10. Second differences: 2, 2, 2. For an² + bn + c, the second difference is 2a, so a = 1.
Subtract n²: 1, 2, 3, 4, 5 remains. Therefore Tn = n² + n.
Recognise 2, 9, 28, 65, 126, …
Compare with n³ = 1, 8, 27, 64, 125. Every term is one larger.
Pause and explainWhy is checking only the first term insufficient?
Many different formulas can give the same first value. A useful proposed rule must match all the given positions.
Exponential and combined sequences
Some sequences grow by repeated multiplication instead of repeated addition. If each term is twice the previous one, powers of 2 describe the growth more naturally than a linear expression.
Write a small table of powers alongside the terms to see whether they differ by a constant, a scale factor or another simple sequence. Be careful about indexing: the first term may use a power of n − 1 rather than n, depending on how the rule starts.
Find the nth term of 3, 6, 12, 24, …
The first term uses exponent 0. Using 3 × 2ⁿ would start at 6.
Simple combinations can be found by comparing with familiar patterns. For 3, 6, 11, 20, 37, … subtract n to get 2, 4, 8, 16, 32. Hence Tn = 2ⁿ + n.
Pause and explainFor 3, 6, 12, 24, why is 3 × 2n − 1 a suitable rule?
At position 1 the power is zero, giving 3. Each next position adds one factor of 2, doubling the term.
Your turn. Check several positions.
Label the positions before searching for a pattern. Test differences for additive patterns and ratios for multiplicative ones.
Once you propose an nth term, substitute the first few positions and at least one later given position. Explain whether the question asks for a next term, a general rule or a particular distant term.
Find the nth term of 7, 12, 17, 22, … and its 50th term.
Reveal worked answer
Find the nth term of 3, 8, 15, 24, 35, …
Reveal worked answer
Continue 5, 15, 45, … and give the nth term.