Geometric Progressions: Concept & Tricks

Introduction
A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. E.g., the height to which a ball rises in each successive bounce follows a geometric progression. The sequence 4, -2, 1,... is a Geometric Progression (GP) for which (-1/2) is the common ratio. We can use the concept to find an arbitrary term, a finite or infinite sum of the series, and apply them in various contexts, including some difficult problems.
  • The general form of a GP is a, ar, ar2, ar3 and so on. Thus nth term of a GP series is Tn = arn-1, where a = first term and r = common ratio = Tm/Tm-1.

If the common ratio is:
  • Negative: the result will alternate between positive and negative.
  • Greater than 1: there will be an exponential growth towards infinity (positive).
  • Less than -1: there will be an exponential growth towards infinity (positive and negative).
  • Between 1 and -1: there will be an exponential decay towards zero.
  • Zero: the result will remain at zero
Geometric Progression Formula
Suppose that we want to find the sum of the first n terms of a geometric progression. What we get is
sum of GP: Sn = a + ar + ar2 + ar3 + . . . + arn-1
  • The GP series formula applied to calculate the sum of geometric progression of first n terms of a GP:
    and
  • When three quantities are in GP, the middle one is called as the geometric mean of the other two. If a, b and c are three quantities in GP and b is the geometric mean of a and c i.e.
  • The sum of infinite GP series: