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Language of Sequences & Series (DP IB Maths: AI HL)
Revision Note
Language of Sequences & Series
What is a sequence?
- A sequence is an ordered set of numbers with a well-defined rule for getting from one number to the next
- For example 1, 3, 5, 7, 9, … is a sequence with the rule ‘start at one and add two to get each subsequent 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 the formula for the nth term
What is a series?
- 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 Sn to refer to the sum of the first n terms in the series
- Sn = u1 + u2 + u3 + … + un
- So for the series above S5 = 1 + 3 + 5 + 7 + 9 = 25
Worked example
Determine the first five terms and the value of S5 in the sequence with terms defined by un = 5 - 2n.
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Sigma Notation
What is 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
- Be careful, the limits don’t have to start with 1
- For example or
- r and k are commonly used variables within sigma notation
Examiner Tip
- Your GDC will be able to use sigma notation, familiarise yourself with it and practice using it to check your work
Worked example
A sequence can be defined by for .
a)
Write an expression for using sigma notation.
b)
Write an expression for using sigma notation.
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