Geometric Progressions (Cambridge O Level Additional Maths)

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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 to the power of t h end exponent term formula for a geometric progression is given as

u subscript n equals a r to the power of n minus 1 end exponent

    • 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 nth term and asked to find the value of n
    • You can solve these using logarithms 

Examiner Tip

  • 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

Geom Seq Example, A Level & AS Level Pure Maths Revision Notes

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 subscript n equals fraction numerator a left parenthesis 1 minus r to the power of n right parenthesis over denominator 1 minus r end fraction   or   S subscript n equals fraction numerator a left parenthesis r to the power of n minus 1 right parenthesis over denominator r minus 1 end fraction
    • is the first term
    • is the common ratio (r not equal to 1)
  • The first is given in the list of formulas and is more convenient if < 1
    • the second is more convenient if > 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 subscript infinity equals fraction numerator a over denominator 1 minus r end fraction

  

  • 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

 

Examiner Tip

  • 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

4-3-2-geom-series-example

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.