1.3.1 Language of Sequences & Series

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 sequence are often called terms
The terms of a sequence are often referred to by letters with a subscript
- In IB this will be the letter u
- So in the sequence above, u₁= 1, u₂= 3, u₃= 5 and so on
Each term in a sequence 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 sequence
- 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

A sequence is increasing if u_n+1> u_nfor all positive integers
- i.e. every term is greater than the term before it
A sequence is decreasing if u_n+1< u_nfor all positive integers
- i.e. every term is less than the term before it
A sequence is periodic if the terms repeat in a cycle

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

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

4-5-2-sigma-not-illustr-1

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

Your GDC will be able to use sigma notation, familiarise yourself with it and practice using it to check your work

A sequence can be defined by $u_{n} = 2 \times 3^{n - 1}$ for $n \in ℤ^{+}$ .

Write an expression for

u_{1} + u_{2} + u_{3} + . . . + u_{6}

using sigma notation.

ai-sl-1-2-1-sigma-a

Write an expression for

u_{7} + u_{8} + u_{9} + . . . + u_{12}

using sigma notation.

ai-sl-1-2-1-sigma-b

DP IB Maths: AI HL

1.3.1 Language of Sequences & Series