H2 Maths 2.1 - Sequences & Series (Foundations)

Step 1 - Sequences and series defined

A sequence is an ordered list of numbers following a rule.

Step 1 - Sequences and series defined

We label the terms u sub one, u sub two, and so on up to u sub n.

Step 1 - Sequences and series defined

A series is what you get when you sum those terms - and two families dominate A-level: arithmetic and geometric progressions.

Step 2 - Arithmetic Progressions

In an arithmetic progression, consecutive terms differ by a fixed common difference d.

Step 2 - Arithmetic Progressions

If the first term is a, the n-th term is a plus n minus one times d.

Step 3 - Sum of an AP

The sum of the first n terms of an arithmetic progression has a closed form.

Step 3 - Sum of an AP

S sub n equals n over two times the bracket two a plus n minus one times d.

Step 3 - Sum of an AP

Equivalently, n over two times the first term plus the last term l.

Step 4 - Geometric Progressions

In a geometric progression, each term is multiplied by a fixed common ratio r.

Step 4 - Geometric Progressions

The n-th term is a times r to the power n minus one.

Step 5 - Sum of a GP and convergence

The sum of the first n terms of a geometric progression is a times r to the n minus one, all over r minus one, for r not equal to one.

Step 5 - Sum of a GP and convergence

When the modulus of r is less than one, the series converges as n tends to infinity.

Step 5 - Sum of a GP and convergence

The sum to infinity is simply a over one minus r.