Language of Sequences & Series (Cambridge International IGCSE Further Maths)

Language of sequences & series

What is a progression?

• 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^thterm

What is a series?

• 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

Worked Example 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. *

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

Final answer: The first five terms are 3, 1, -1, -3, -5

To find S₅ add the first five terms of the progression together.

3+1+(−1)+(−3)+(−5)

Final answer: S₅ = -5

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 $\sum_{k=0}^{4} ( 2k+1 )$ or $\sum_{k=7}^{14} ( 2k−13 )$
  • r and k are commonly used variables within sigma notation

Exam Hint

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