Geometric Series (CIE A Level Maths: Pure 1)

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Geometric Series

How do I find the sum of a geometric progression?

  • The sum of the terms of a geometric progression is sometimes called a geometric series

 Geom Series Illustr, A Level & AS Level Pure Maths Revision Notes 

  • The following formulae will let you find the sum of the first n terms of a geometric progression:
begin mathsize 22px style 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 end style   or   begin mathsize 22px style 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 end style
    • is the first term
    • is the common ratio

  

  • The one on the left is more convenient if < 1, the one on the right is more convenient if > 1
  • The a and the r in those formulae are exactly the same as the ones used with geometric progression

 

How do I prove the formula for the sum of a geometric progression?

  • Learn this proof of the sum of a geometric progression formula – you can be asked to give it in the exam:
    • Write out the sum once
    • Write out the sum again but multiply each term by r
    • Subtract the second sum from the first
      • All the terms except the two should cancel out
    • Factorise and rearrange to make the subject

Geom Series Proof, A Level & AS Level Pure Maths Revision Notes 

What is the sum to infinity of a geometric series?

  • If (and only if!) |r| < 1, then the sum of a geometric progression converges to a finite value given by the formula
begin mathsize 22px style S subscript infinity equals fraction numerator a over denominator 1 minus r end fraction end style

  

  • 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

 

Worked example

4-3-2-geom-series-example

Examiner Tip

The geometric series formulae are in the formulae booklet – you don't need to memorise them. 

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.