Sequences and series

A sequence is a list of numbers in a definite order: a first term a₁, a second a₂, a third a₃, and so on. The nth term aₙ is also called the general term, because it gives any term you ask for.

• 2, 5, 8, 11, ... has aₙ = 3n – 1, so a₁₀ = 3 × 10 – 1 = 29.
• aₙ = (–1)ⁿ × n gives –1, 2, –3, 4, ...: the signs take turns.
• A sequence can stop (finite: the months of a year) or go on for ever (infinite: all odd numbers).

Not every sequence has a neat formula. The prime numbers 2, 3, 5, 7, 11, ... follow a clear rule, but there is no simple formula for the nth prime.

Some sequences are given by a recurrence: a rule that builds each term from the ones before it.

• a₁ = 2 and aₙ = 2aₙ₋₁ – 1: a₂ = 2 × 2 – 1 = 3, a₃ = 5, a₄ = 9, a₅ = 17.
• The Fibonacci sequence starts a₁ = a₂ = 1 and adds the two previous terms: 1, 1, 2, 3, 5, 8, 13, 21, ... The ratio of neighbours settles down near 1.618: 13/8 = 1.625, 21/13 ≈ 1.615.

Because each position 1, 2, 3, ... gets exactly one value, a sequence is really a function whose inputs are the natural numbers.

A series is what you get when you add the terms of a sequence: a₁ + a₂ + a₃ + ... The Greek letter Σ (sigma) means 'add up'.

• Σ (k = 1 to 4) k² means put k = 1, 2, 3, 4 into k² and add: 1 + 4 + 9 + 16 = 30.

From Class 10: the A.P. In an arithmetic progression each term is the previous one plus a fixed number d (the common difference): aₙ = a + (n – 1)d and Sₙ = (n/2)[2a + (n – 1)d] = (n/2)(first + last).

• The arithmetic mean of a and b is (a + b)/2, and it makes a, (a + b)/2, b an A.P.
• To put n numbers between a and b so that the whole list is an A.P., there are n + 2 terms in all, so d = (b – a)/(n + 1). Between 2 and 14, three numbers: d = 12/4 = 3, giving 5, 8, 11.
• Three numbers a, b, c are in A.P. exactly when 2b = a + c.