Sum of Geometric Progressions (Cambridge International AS Mathematics)

Geometric Series

Sum of geometric progressions

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

  • *a *is the first term
  • *r *is the common ratio

• 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 progression?

• If (and only 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

Worked Example

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