4.3.2 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:

• • 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