Arithmetic Sequences & Series (DP IB Maths: AI HL)

What is an arithmetic sequence?

• In an arithmetic sequence, the difference between consecutive terms in the sequence is constant
• This constant difference is known as the common difference, d, of the sequence
  • For example, 1, 4, 7, 10, … is an arithmetic sequence with the rule ‘start at one and add three to each number’
    • The first term, u₁, is 1
    • The common difference, d, is 3
  • An arithmetic sequence can be increasing (positive common difference) or decreasing (negative common difference)
  • Each term of an arithmetic sequence is referred to by the letter u with a subscript determining its place in the sequence

How do I find a term in an arithmetic sequence?

• The n^th term formula for an arithmetic sequence is given as

• • Where is the first term, and is the common difference
  • This is given in the formula booklet, you do not need to know how to derive it
• Enter the information you have into the formula and use your GDC to find the value of the term
• Sometimes you will be given a term and asked to find the first term or the common difference
  • Substitute the information into the formula and solve the equation
    • You could use your GDC for this
• Sometimes you will be given two terms and asked to find both the first term and the common difference
  • Substitute the information into the formula and set up a system of linear equations
  • Solve the simultaneous equations
    • You could use your GDC for this

How do I find the sum of an arithmetic series?

• An arithmetic series is the sum of the terms in an arithmetic sequence
  • For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …
• Use the following formulae to find the sum of the first n terms of the arithmetic series:

• • • is the first term
    • is the common difference
    • is the last term
  • Both formulae are given in the formula booklet, you do not need to know how to derive them
• You can use whichever formula is more convenient for a given question
  • If you know the first term and common difference use the first version
  • If you know the first and last term then the second version is easier to use
• A question will often give you the sum of a certain number of terms and ask you to find the value of the first term or the common difference
  • Substitute the information into the formula and solve the equation
    • You could use your GDC for this