Language of Sequences & Series (DP IB Maths: AI HL)

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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.

ai-sl-1-2-1-language-we-so

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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

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

  • Be careful, the limits don’t have to start with 1
    • For example sum from k space equals space 0 to 4 of left parenthesis 2 k plus 1 right parenthesis  or  sum from k space equals space 7 to 14 of left parenthesis 2 k minus 13 right parenthesis
    • 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  u subscript n equals space 2 space cross times space 3 to the power of n minus 1 end exponent for  n element of space straight integer numbers to the power of plus .

 

a)
Write an expression for u subscript 1 space plus space u subscript 2 space plus space u subscript 3 space plus space... space plus space u subscript 6 using sigma notation.

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

 

b)
Write an expression for u subscript 7 space plus space u subscript 8 space plus space u subscript 9 space plus space... space plus space u subscript 12using sigma notation.

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

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.