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Geometric Series (CIE A Level Maths: Pure 1)
Revision Note
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
- The following formulae will let you find the sum of the first n terms of a geometric progression:
or
-
- a is the first term
- r is the common ratio
- The one on the left is more convenient if r < 1, the one on the right is more convenient if r > 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 S the subject
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
- 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
Examiner Tip
The geometric series formulae are in the formulae booklet – you don't need to memorise them.
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