GCSE Maths · Sequences and nth Term · Foundation · Summary
GCSE Maths Sequences and nth Term (Foundation): summary and key words
Part of the GCSE Maths sequences and nth term chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page
Chapter summary
| Skill | Method |
|---|---|
| Term-to-term rule | Find the differences (add/subtract), then the ratios (multiply/divide); apply the rule to the last term |
| Position-to-term rule | Replace n by the position and follow BIDMAS; the first term is n = 1 |
| Pattern sequences | Count each pattern, find the rule, explain it with the picture, continue the sequence |
| nth term of an arithmetic sequence | Find d, write the d times table, subtract to find the adjustment: dn + (first term − d) |
| Is a number in the sequence? | Solve nth term = number; it is a term only if n is a positive whole number |
| First term above (below) a value | Solve an inequality; round n up (down) to a whole number and test the term before |
| Special sequences | Squares n², cubes n³, triangular n(n + 1) ÷ 2, powers of 2: 2n, Fibonacci-type: add the previous two |
| Geometric sequences | Common ratio r = term ÷ previous term; powers rn; Higher: nth term arn − 1; surds: √k × √k = k |
| Two terms given (Higher) | d = (change in terms) ÷ (change in positions), then find the adjustment |
| Quadratic sequences (Higher) | Second differences constant; a = half the second difference; subtract an²; find the linear part bn + c |
| Proving with sequences | Use the nth term and algebra, e.g. 6n + 3 = 3(2n + 1) |
Can you…?
- I can continue a sequence and describe its term-to-term rule in words.
- I can find terms from a position-to-term rule, including powers such as n² and 2n.
- I can continue patterns made from matchsticks, tiles and dots.
- I can find the nth term of an arithmetic sequence.
- I can decide whether a number is in a sequence and find the first term above or below a value.
- I can recognise square, cube, triangular, Fibonacci-type and geometric sequences.
- I can find the common ratio of a geometric sequence and continue it.
- Higher: I can find the nth term of a quadratic sequence using second differences.
- Higher: I can find the nth term of a geometric sequence and work with a surd ratio.
- I can use the nth term to explain or prove something about every term.
Key terms
- Sequence
- A list of numbers that follow a rule.
- Term
- One number in a sequence.
- Term-to-term rule
- A rule that tells you how to get from one term to the next.
- Position-to-term rule (nth term)
- A formula that gives the term from its position n.
- Position
- The place of a term in the sequence: 1st, 2nd, 3rd, …
- Arithmetic (linear) sequence
- A sequence that goes up or down by the same amount each time.
- Common difference
- The amount added each time in an arithmetic sequence.
- Geometric sequence
- A sequence made by multiplying by the same number each time.
- Common ratio
- The number multiplied each time in a geometric sequence.
- Square number
- A number multiplied by itself: 1, 4, 9, 16, …
- Cube number
- A number multiplied by itself twice more: 1, 8, 27, 64, …
- Triangular number
- The numbers 1, 3, 6, 10, 15, … found by adding 1, then 2, then 3, …
- Fibonacci-type sequence
- A sequence where each term is the sum of the two terms before it.
- First differences
- The differences between neighbouring terms.
- Second differences
- The differences between the first differences.
- Quadratic sequence
- A sequence whose nth term contains n²; its second differences are constant (Higher).
- Surd
- A root that is not exact, such as √2 (Higher).