What is a sequence?

• 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,?u_1?= 1,?u_2?= 3,?u_3?= 5 and so on
• Each term in a sequence can be found by?substituting?the term number into?formula for the n^th?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?S_n?to refer to the sum of the first?n?terms in the series
  • S_n?= u_1?+?u_2?+?u_3?+ … +?u_n
  • So for the series above?S₅?=?1 + 3 + 5 + 7 + 9 = 25

What are increasing, decreasing and periodic sequences?

• A sequence is?increasing?if?u_n+1?> u_n?for all positive integers
  • i.e. every term is greater than the term before it
• A sequence is?decreasing?if?u_n+1?< u_n?for 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

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