Sequences: arithmetic, geometric, Fibonacci and the nth term
Arithmetic sequences
An arithmetic sequence has a constant common difference (d) between consecutive terms.
General term (nth term): aₙ = a + (n − 1)d
Where a = first term, d = common difference.
Example: 5, 8, 11, 14, … → a = 5, d = 3. nth term = 5 + (n − 1)(3) = 5 + 3n − 3 = 3n + 2. Check: n = 1 → 5 ✓; n = 2 → 8 ✓.
Finding d and a from two terms: if aₘ and aₙ are known, solve simultaneous equations.
Sum of arithmetic sequence (Higher): Sₙ = n/2 × (first term + last term) = n/2 × (2a + (n − 1)d).
Geometric sequences
A geometric sequence has a constant common ratio (r) between consecutive terms.
General term: aₙ = a × rⁿ⁻¹
Example: 3, 6, 12, 24, … → a = 3, r = 2. nth term = 3 × 2ⁿ⁻¹.
Test: divide any term by the previous — if the ratio is constant, it's geometric.
Convergence (Higher): if |r| < 1, the sequence converges to 0 as n → ∞. Sum to infinity (|r| < 1): S∞ = a / (1 − r).
Fibonacci-type sequences
Each term is the sum of the two previous terms: 1, 1, 2, 3, 5, 8, 13, …
Variations: start with different values (e.g. a, b, a+b, a+2b, 2a+3b…). CCEA may give the first two terms and ask you to find the nth term or a particular value using the Fibonacci rule.
Finding the nth term from a pattern
For linear sequences (arithmetic):
- Find d (difference between consecutive terms).
- nth term = dn + c, where c = (first term − d).
For quadratic sequences (second differences are constant):
- The leading coefficient is (second difference)/2, giving an² term.
- Subtract the quadratic part to find a linear/constant adjustment.
Example: 3, 8, 15, 24, 35, … First differences: 5, 7, 9, 11 → second differences: 2, 2, 2 (constant → quadratic). Leading term: 2/2 = 1 → n² term. Sequence of n²: 1, 4, 9, 16, 25. Subtract: 3−1=2, 8−4=4, 15−9=6, 24−16=8 → 2n term. nth term = n² + 2n.
CCEA context
CCEA Paper 1: find the nth term of an arithmetic sequence, extend a Fibonacci-type sequence, identify the type of sequence. CCEA Paper 2: geometric sequences, sum formulas, quadratic nth terms.
⚠Common mistakes
- Arithmetic nth term: writing dn + a instead of d(n−1) + a = dn + (a − d).
- Geometric sequence: using the common difference method instead of ratio.
- Quadratic sequences: only finding the quadratic term and forgetting the linear/constant adjustment.
- Fibonacci: forgetting the sequence rule; calculating aₙ = aₙ₋₁ + aₙ₋₂ not aₙ₋₁ + aₙ₋₃.
AI-generated · claude-opus-4-7 · v3-ccea-maths