Arithmetic Progressions (Cambridge International IGCSE Further Maths)

Arithmetic Sequences

What is an arithmetic progression?

• In an arithmetic progression(also called 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, a, is 1
  • The common difference, d, is 3

• An arithmetic progression can be increasing (positive common difference) or decreasing(negative common difference)

• Each term of an arithmetic progression 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 progression?

• The n^thterm formula for an arithmetic progression is given as

$u_{n}=a+(n-1)d$

• Where $a$* *is the first term, and $d$ is the common difference

  • This is given on the list of formulas page of the exam, you do not need to know how to derive it

• 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

• 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 two simultaneous equations
  • Solve the simultaneous equations

Exam Hint

• Simultaneous equations are often needed within arithmetic progression questions

  • Make sure you are confident solving them

Worked Example

Arithmetic Series

What is an arithmetic series?

• An arithmetic series is the sum of the terms in an arithmetic progression

  • It is often referred to as the sum of an arithmetic progression
  • For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …

How do I find the sum of an arithmetic progression?

• Use the following formulae to find the sum of the first n terms of the arithmetic series:

$S_ n=1 \frac{1}{2} n \{ a+l \}=1 \frac{1}{2} n \{ 2a+( n−1 ) d \}$* *

• $a$_{* *}is the first term

  • $l$_{* *}is the last term
  • $d$ is the common difference
  • $n$_{* *}is the number of terms in the series
  • Both formulae are given on the formula page, 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 second version
  • If you know the first and last term then the first 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

Exam Hint

• The arithmetic series formulae are given on page 2 of the exam paper – you don't need to memorise them

  • Practise finding the formulae so that you can quickly locate them in the exam

Worked Example