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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p^0, p^1, p^2, p^3,...)}
is of some interest (vide: we are starting with the exponent zero here.) This sequence would be:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p^0, p^0+p^1, p^0+p^1+p^2, p^0+p^1+p^2+p^3, ...)}
and is called a Geometric Series.
How to calculate this? Now, if we look at the n-th element of this sequence, we see:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p-1) \cdot (p^0 + p^1 + p^2 + ... + p^n) }
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =p^1 + p^2 + p^3 + ... + p^{n+1}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - p^0 - p^1 - p^2 - ... - p^n}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =p^{n+1}-p^0}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =p^{n+1}-1}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftrightarrow}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^0+p^1+p^2+...+p^n = \frac{p^{n+1}-1}{p-1}}
Obviously, the last step is allowed only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \neq 1 }
. So, the sequence of partial sums is (if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \neq 1 }
):
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{p-1} (p^1-1, p^2-1,p^3-1, ...)} - and it will converge for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 < p < 1 } to the limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{1-p}} .