Language of Sequences & Series

A progression (also called a sequence) is an ordered set of numbers with a rule for finding all of the numbers in the sequence
- For example 1, 3, 5, 7, 9, … is a sequence with the rule ‘start at one and add two to each number’
The numbers in a progression are often called terms
The terms of a progression are often referred to by letters with a subscript
- This will often be the letter u
- So in the progression above, u₁= 1, u₂= 3, u₃= 5 and so on
Each term in a progression can be found by substituting the term number into formula for the n^th term

You get a series by summing up the terms in a progression
- E.g. For the sequence 1, 3, 5, 7, … the associated series is 1 + 3 + 5 + 7 + …
We use the notation S_n to refer to the sum of the first n terms in the series
- S_n = u₁+ u₂+ u₃+ … + u_n
- So for the series above S₅ = 1 + 3 + 5 + 7 + 9 = 25

Determine the first five terms and the value of S₅ in the progression with terms defined by u_n = 5 - 2n.

Substitute n in for each term that you want to find.

$u_{1} = 5 - 2 (1) = 3 u_{2} = 5 - 2 (2) = 1 u_{3} = 5 - 2 (3) = - 1 u_{4} = 5 - 2 (4) = - 3 u_{5} = 5 - 2 (5) = - 5$

The first five terms are 3, 1, -1, -3, -5

To find S₅add the first five terms of the progression together.

$3 + 1 + (- 1) + (- 3) + (- 5)$

S₅= -5

Sigma Notation

Sigma notation is used to show the sum of a certain number of terms in a sequence
The symbol Σ is the capital Greek letter sigma
Σ stands for ‘sum’
- The expression to the right of the Σ tells you what is being summed, and the limits above and below tell you which terms you are summing

Sigma notation explained

Be careful, the limits don’t have to start with 1
- For example $\sum_{k = 0}^{4} (2 k + 1)$ or $\sum_{k = 7}^{14} (2 k - 13)$
- r and k are commonly used variables within sigma notation

Sigma notation will not be tested in the exam but understanding it will help you to further understand series