5 min readConcept sheetPart I (1 mark) and Part II (2 marks) to find given terms or the general term

Sequences

A sequence is a list of numbers in order: a₁, a₂, a₃, .... It is a function from N to R, and aₙ is its general term.

SequenceA real valued function on the natural numbers: position n gives the term aₙ

General termaₙ, the nth term: a rule that gives the term at any position n

Finite or infiniteA finite sequence has a last term; an infinite sequence goes on forever

Recursive ruleTerms built from earlier terms, e.g. a₁ = a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3 (Fibonacci)

A sequence is a function

In a sequence, every position n = 1, 2, 3, ... has exactly one term aₙ. So a sequence is a function whose domain is the natural numbers N. Terms may repeat, but the order matters: 1, 2, 3 and 3, 2, 1 are different sequences. A finite sequence uses only the positions 1, 2, ..., n.

Words to know

Term: each number in the sequence; a₁ is the first term, a₂ the second, and so on.
General term or nth term aₙ: a formula in n that gives every term.
Finite sequence: it has a last term. Infinite sequence: it goes on without end, shown by '...'.
Some sequences are given by a recursive rule that uses earlier terms, such as the Fibonacci sequence 1, 1, 2, 3, 5, 8, ....
Alternating signs come from (−1)ⁿ (starts with −) or (−1)ⁿ⁺¹ (starts with +).

Example 1: terms from the general term

Problem

Find the first three terms and the 10th term of the sequence aₙ = (2n + 1)/(n + 2).

Working

n = 1: a₁ = (2 + 1)/(1 + 2) = 3/3 = 1
n = 2: a₂ = (4 + 1)/(2 + 2) = 5/4
n = 3: a₃ = (6 + 1)/(3 + 2) = 7/5
n = 10: a₁₀ = (20 + 1)/(10 + 2) = 21/12 = 7/4

Answer

a₁ = 1, a₂ = 5/4, a₃ = 7/5 and a₁₀ = 7/4.

Example 2: the general term from a pattern

Problem

Find the general term of 3, 8, 15, 24, 35, ... and write the next term.

Working

Compare each term with its position: 3 = 1 × 3, 8 = 2 × 4, 15 = 3 × 5, 24 = 4 × 6.
The term at position n is n × (n + 2).
So aₙ = n(n + 2) = n² + 2n.
Check: a₅ = 5 × 7 = 35, which matches.
Next term: a₆ = 6 × 8 = 48.

Answer

aₙ = n(n + 2); the next term is 48.

Example 3: a recursive sequence

Problem

A sequence has a₁ = 2 and aₙ = 3aₙ₋₁ − 1 for n ≥ 2. Find its first five terms.

Working

a₁ = 2
a₂ = 3a₁ − 1 = 3(2) − 1 = 5
a₃ = 3a₂ − 1 = 3(5) − 1 = 14
a₄ = 3a₃ − 1 = 3(14) − 1 = 41
a₅ = 3a₄ − 1 = 3(41) − 1 = 122

Answer

2, 5, 14, 41, 122

In real life

Token numbers at a ration shop or a hospital OP counter form a sequence: positions 1, 2, 3, ... each get exactly one number, in order. A step count noted every day is also a sequence.

Exam angle

2 marks: “Find a₈ and a₁₅ for the sequence aₙ = (−1)ⁿ (n + 2)/n.” a₈ = (+1) × 10/8 = 5/4. a₁₅ = (−1) × 17/15 = −17/15.

2 marks: “Find the next three terms of 1, 1, 2, 3, 5, 8, ... and state the rule.” Each term is the sum of the two before it, so the next terms are 13, 21 and 34. Rule: a₁ = a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3.

Remember it

Position in, term out: a sequence is a function from 1, 2, 3, ... to the terms.