Sequences, nth term and recurrence
Use aₙ notation to find terms, build a sequence from a recurrence relation, spot a periodic sequence and count triangular numbers.
- UK: Year 12
- France: grade 11
- Iran: grade 10
- Afghanistan: grade 11
- Syria: grade 11
- Palestine: grade 11
- Ukraine: grade 9
- Russia: grade 9
Guide
A sequence is a list of numbers: a₁ is the first term, a₂ the second and aₙ the nth. If aₙ = 3n − 2, we put the term's position in place of n: a₄ = 3×4 − 2 = 10.
Sometimes each term is built from the one before. That is a recurrence relation: a₁ = 2 and aₙ₊₁ = 3aₙ − 1 means the next term is three times the current one, minus 1. We build the terms one by one.
A sequence that goes up is increasing; one that goes down is decreasing. One that keeps repeating the same numbers is periodic: 5, −5, 5, −5 …
The triangular numbers 1, 3, 6, 10, … come from adding 1 + 2 + 3 + … . The Fibonacci sequence 1, 1, 2, 3, 5, 8 … takes each term from the sum of the two before.
Common slips: changing the sign of the constant; missing one step in a recurrence; swapping odd and even positions; confusing triangular numbers with square numbers.