Geometric Progressions (Cambridge International IGCSE Further Maths)

Geometric Sequences

What is a geometric progression?

• In a geometric progression (also called geometric sequence) there is a common ratio, r, between consecutive terms in the sequence

  • For example, 2, 6, 18, 54, 162, … is a progression with the rule ‘start at two and multiply each number by three’

    • The first term, a, is 2
    • The common ratio, r, is 3

• A geometric progression can be increasing (r > 1) or decreasing(0 < r < 1)

• If the common ratio is a negative number the terms will alternate between positive and negative values

  • For example, 1, -4, 16, -64, 256, … is a sequence with the rule ‘start at one and multiply each number by negative four’

    • The first term, a, is 1

      • The common ratio, r, is -4

• Each term of a geometric progression is referred to by the letter u with a subscript determining its place in the sequence

How do I find a term in a geometric progression?

• The $n^{th}$ term formula for a geometric progression is given as

$u_{n}=ar^{n-1}$

• Where $a$* *is the first term, and $r$ is the common ratio

  • This formula allows you to find any term in the geometric progression
  • It is given in the list of formulas, you do not need to know how to derive it

• Enter the information you have into the formula and find the value of the term

• Sometimes you will be given a term and asked to find the first term or the common ratio

  • Substitute the information into the formula and solve the equation

• Sometimes you will be given two or more consecutive terms and asked to find both the first term and the common ratio

  • Find the common ratio by dividing a term by the one before it
  • Substitute this and one of the terms into the formula to find the first term

• Sometimes you may be given a term and the formula for the n^th term and asked to find the value of n

  • You can solve these using logarithms

Exam Hint

• The formula is given in the list of formulas
• If you know two terms in a geometric progression you can find a and r using simultaneous equations

Worked Example

Geometric Series

What is a geometric series?

• A geometric series is the sum of the terms in a geometric progression

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

How do I find the sum of a geometric progression?

• The following formulae will let you find the sum of the first n terms of a geometric progression:

$S_{n}=\frac{a(1-r^{n})}{1-r}$ or $S_{n}=\frac{a(r^{n}-1)}{r-1}$

• a is the first term

  • r is the common ratio ($r\neq1$)

• The first is given in the list of formulas and is more convenient if r < 1

  • the second is more convenient if r > 1

• 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, the common ratio, or the number of terms within the sequence

  • Substitute the information into the formula and solve the equation

What is the sum to infinity of a geometric series?

• If |r| < 1, then the sum of a geometric progression converges to a finite value given by the formula

$S_{\infty}=\frac{a}{1-r}$

• S_∞ is known as the sum to infinity
• If |r| ≥ 1 the sum of a geometric progression is divergent and the sum to infinity does not exist

Exam Hint

• The geometric series formulae are in the list of formulas on page 2 – you don't need to memorise them

  • Make sure you can locate them quickly before going into the exam

Worked Example