Geometric Progression Formulas

In mathematics, a geometric progression (also inaccurately known as a geometric series) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.

a geometric sequence can be written as:

aq0=a, aq1=aq, aq2, q3, ... where q ≠ 0, q is the common ratio and a is a scale factor.

Formulae for the n-th term can be defined as:

an = an-1.q
an = a1.qn-1

The common ratio then is:

q =
ak
ak-1

A sequence with a common ratio of 2 and a scale factor of 1 is
1, 2, 4, 8, 16, 32...

A sequence with a common ratio of -1 and a scale factor of 5 is
5, -5, 5, -5, 5, -5,...

If the common ratio is:

  • Negative, the results will alternate between positive and negative.
  • Greater than 1, there will be exponential growth towards infinity (positive).
  • Less than -1, there will be exponential growth towards infinity (positive and negative).
  • Between 1 and -1, there will be exponential decay towards zero.
  • Zero, the results will remain at zero

Geometric Progression Properties

a2k = ak-1.ak+1
a1.an = a2.an-1 =...= ak.an-k+1

Formula for the sum of the first n numbers of a geometric series

Sn = a1 - anq 1  -  q  = a1. 1 - qn 1 - q

Infinite geometric series where |q| < 1


If |q| < 1 then an -> 0, when n -> ∞.
The sum S of such an infinite geometric series is given by the formula:

S = a1
1
1 - q
which is valid only when |q| < 1.
a1 is the first term.

Geometric Progression Calculator

First term
Common ratio
Number of terms(n=?)

Geometric Progression Problems

Problem 1.
Is the sequence 2, 4, 6, 8... a geometric progression?
Solution: No, it is not. (2, 4, 8 is a geometric progression)


Problem 2
If 2, 4, 8... form a geometric progression. What is the 10-th term?
Solution: We can use the formula an = a1 . qn-1
a10 = 2 . 210-1 = 2 . 512 = 1024


Problem 3
Find the scale factor and the command ratio of a geometric progression if
a5 - a1 = 15
a4 - a2 = 6
Solution: there are two geometric progressions. The first one has a scale factor 1 and common ratio = 2
the second decidion is -16, 1/2

Additional problems:
Geometric progression problems
Progression problems

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