Sequences

• Sequences are ordered lists of numbers (terms) that follow a rule, and each term has a position, labelled with n or as a subscript like an.
• Term-to-term rules tell you how to get from one term to the next (add, subtract, multiply, divide).
• Position-to-term rules (the nth term formula) let you jump directly to any term without building the whole sequence.
• Linear sequences have a constant difference between terms — formula is dn + b.
• Quadratic sequences have a constant second difference — formula is an² + bn + c.
• Cubic sequences have a constant third difference — formula is an³ + bn² + cn + d.
• Exponential (geometric) sequences multiply by the same number each time — formula is a × rⁿ.
• You can check if a value belongs to a sequence by setting the formula equal to that value and solving for n — if n is a positive whole number, it's in the sequence.
• Harder problems combine sequences (fractions of two sequences, sums of two sequences) or require solving simultaneous equations.

A sequence is simply an ordered set of numbers that follow a rule. For example: 3, 6, 9, 12, ... — here the rule is "add 3 each time."

Each number in the list is called a term. The position of a term tells you where it sits in the list — the letter n is used for this. So:

• n = 1 refers to the 1st term
• n = 2 refers to the 2nd term
• If you don't know the exact position, you just call it the "nth term"

You'll also see sequences written using subscripts: a general sequence is a₁, a₂, a₃, ... where a₁ is the 1st term, a₂ is the 2nd term, and aₙ represents the nth term (any term you like).

Term-to-term rules tell you how to get the next term from the term you're currently on. It's the operation you repeat each time. Common ones include:

• Adding or subtracting a number — e.g. 3, 7, 11, 15, ... (add 4 each time)
• Multiplying or dividing by a number — e.g. 2, 6, 18, 54, ... (multiply by 3 each time)

The catch with term-to-term rules: to find the 50th term, you'd have to work out all 49 terms before it! That's where position-to-term rules save the day.

A position-to-term rule — also called the nth term formula — is an algebraic expression in n that lets you calculate any term directly, just by plugging in its position number. No need to work out every term before it!

n