Geometric progression

A geometric progression is a sequence of numbers that has a constant ratio of each term to its preceding term. For example, this is a geometric progression: 2, 4, 8, 16, 32.

In finance, compound interest is an example of a geometric progression. An example would be a bank account that earns an API (annual percentage interest) rate of 5% per year. Every year, the number of dollars in the account is multiplied by the factor 1.05. That is about the same as doubling every fifteen years.

The sequence 2, 4, 8, 16, 32 is simply the "powers of two:" two, two squared, two cubed, two to the fourth power, and so on. This can be written using exponents this way:

2¹, 2², 2³, 2⁴, 2⁵

This is why geometric progressions are sometimes called exponential growth.

Real-life examples

Moore's Law is the statement that the number of transistors in common integrated circuits roughly doubles every two years.

The growth of new conservative words follows (approximately) a geometric progression. The number of conservative words roughly doubles every century.

Geometric series

Often, the sequence of partial sums of a geometric progression <math>(p^0, p^1, p^2, p^3,...)</math> is of some interest (vide: we are starting with the exponent zero here.) This sequence would be:

<math>(p^0, p^0+p^1, p^0+p^1+p^2, p^0+p^1+p^2+p^3, ...)</math> and is called a Geometric Series.
How to calculate this? Now, if we look at the n-th element of this sequence, we see:

• <math>(p-1) \cdot (p^0 + p^1 + p^2 + ... + p^n) </math>
• <math>=p^1 + p^2 + p^3 + ... + p^{n+1}</math><math> - p^0 - p^1 - p^2 - ... - p^n</math>
• <math>=p^{n+1}-p^0</math>
• <math>=p^{n+1}-1</math>
• <math>\Leftrightarrow</math>
• <math>p^0+p^1+p^2+...+p^n = \frac{p^{n+1}-1}{p-1}</math>

Obviously, the last step is allowed only if <math>p \neq 1 </math>. So, the sequence of partial sums is (if <math>p \neq 1 </math>):

<math>\frac{1}{p-1} (p^1-1, p^2-1,p^3-1, ...)</math> - and it will converge for <math>-1 < p < 1 </math> to the limit <math>\frac{1}{1-p}</math>.

See also

• Arithmetic progression