Sequences and Series

What is a sequence?

A sequence is made up of terms in an order. Terms in a sequence can be numbers or algebraic expressions. Some examples include:
1, 2, 3, 4, 5, 6, …
1, 1, 1, 1, 1, 1, …
1, 2, 1, 2, 1, 2, …
1, 2, 4, 8, 16, 32, …
1, 3, 5, 7, 9, 11, …
a, b, b, a, b, b, a, …

What is a progression?

The word “progression” in maths is just another word for a sequence.

What is the difference between a term-to-term rule and a position-to-term rule?

Sometimes a sequence might be defined by a rule. The rule could be given as term-to-term or position-to-term. The term-to-term rule is also known as the recursion rule. A term-to-term rule describes how to get from previous terms to the next. The rule must also state how the sequence must be started. The sequence 1, 2, 4, 8, 16, … has the term-to-term rule of “double the previous term and start with 1”. Whereas 2, 4, 8, 16, 32, … has a similar rule but starts with 2 instead of 1. The sequence 1, 2, 3, 5, 8, 13, … has the rule “add together the previous two terms and start with 1, 2”.

A position-to-term rule describes how to find any term in a sequence given its position. The rule does not need to state how the sequence starts. The sequence “2, 4, 6, 8, 10, …” has the position-to-term rule of “double the position of the term”. The sequence “4, 9, 16, 25, 36, …” has the rule “add 1 to the position of the term and then square it”.

Term-to-term rules tend to be easier to spot but position-to-term rules tend to be more useful.

What notation is used for sequences?

Terms in a sequence are usually denoted with a lowercase letter and a subscript denoted the position of the term, an and un are the most commonly used expressions. Sequences can also be denoted using function notation such as T(n) where T(3) would mean the 3rd term in the sequence.

How do I find the term-to-term rule?

Step 1: Spot the pattern. Look for whether the terms are being multiplied are added to a constant term.
Step 2: Write down the rule using algebra. This is commonly done by writing u_n+1 in terms of u_n and/or previous terms.
Step 3: Write down the first term(s) that are needed to define the sequence.

Examples:
1, 2, 4, 8, 16, … u_n+1 = 2u_n and u₁ = 1
1, 2, 3, 5, 8, 13, … u_n+1 = u_n + u_n-1 and u₁ = 1, u₂ = 2 

How do I find the position-to-term rule?

Step 1: Find the rule that connects the position of a term to the term itself. This can be done by identifying the type of sequence and/or looking at the differences between the terms.
Step 2: Write down the rule using algebra. This is commonly done by writing u_n is terms of n.

Examples:
2, 4, 6, 8, 10, … u_n = 2n
4, 9, 16, 25, 36, … u_n = (n + 1)²

Does a sequence always have a term-to-term rule or a position-to-term rule?

No! Sequences could be anything. There could be sequences which has a simple term-to-term rule but a complicated position-to-term rule. The Fibonacci sequence (1,1,2,3,5,8, …) is created starting with two 1s and then finding each term by adding together the previous two terms. However the position-to-term rule is a lot more complicated to find:

What is an arithmetic sequence?

An arithmetic sequence is one where the term-to-term rule is “add a given constant to the previous term”. The constant is called the common difference and is usually denoted by the letter d. The first term is usually denoted by the letter a. The term-to-term rule is u_n+1 = u_n + d, u₁=a. The position-to-term rule is u_n = a + (n-1)d.

What is a geometric sequence?

A geometric sequence is one where the term-to-term rule is “multiply the previous term by a given constant”. The constant is called the common ratio and is usually denoted by the letter r. The first term is usually denoted by the letter a. The term-to-term rule is u_n+1 = ru_n, u₁=a. The position-to-term rule is u_n = ar^n-1.

What does it mean if a sequence converges?

A sequence converges if the terms get closer and and closer to a fixed limit. For example,0.3, 0.33, 0.333, 0.3333, … converges to ⅓. Geometric sequences will converge if the common ratio is between -1 and 1.

What is a series?

A series is the sum of the first n terms of a sequence. It is common to use the notation S_n to denote the sum of the first n terms. There are formulae for this if the sequence is arithmetic or geometric.

Arithmetic: 
Geometric: 

The notation  means the sum of terms of the sequence u_n starting with position a and ending with position b.

What is a convergent series?

A series converges if the sum tends to a fixed limit as you include more terms in the series. If a series is convergent then the terms in the sequence must converge to zero. For example, 0.3+0.03+0.003+...converges to ⅓ and the terms in the sequence converge to zero. The converse is not true. If a sequence converges to zero then the series must not converge. ½, ⅓, ¼, ⅕, … converges to 0 however ½+⅓+¼+⅕+... tends to infinity.

What is an arithmetic series?

An arithmetic series is a series where the sequence is arithmetic. An arithmetic series is only convergent if the common difference is 0.

What is a geometric series?

A geometric series is a series where the sequence is geometric. A geometric series is only convergent if the common ratio is between -1 and 1. The limit (or sum to infinity) is