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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.

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.

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:

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 [[exponent]]s this way:

::2¹, 2², 2³, 2⁴, 2⁵

::2¹, 2², 2³, 2⁴, 2⁵

This is why geometric progressions are sometimes called "exponential growth."

This is why geometric progressions are sometimes called [[exponential growth]].

 

==Geometric Series==

==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: <br />

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: <br />

<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'''. <br>How to calculate this? Now, if we look at the n-th element of this sequence, we see: <ul><li><math>(p-1) \cdot (p^0 + p^1 + p^2 + ... + p^n) </math><li><math>=p^1 + p^2 + p^3 + ... + p^{n+1}</math><math> - p^0 - p^1 - p^2 - ... - p^n</math><li><math>=p^{n+1}-p^0</math><li><math>=p^{n+1}-1</math><li><math>\Leftrightarrow</math><li><math>p^0+p^1+p^2+...+p^n = \frac{p^{n+1}-1}{p-1}</math></ul>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>):<br>

<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'''. <br>How to calculate this? Now, if we look at the n-th element of this sequence, we see: <ul><li><math>(p-1) \cdot (p^0 + p^1 + p^2 + ... + p^n) </math><li><math>=p^1 + p^2 + p^3 + ... + p^{n+1}</math><math> - p^0 - p^1 - p^2 - ... - p^n</math><li><math>=p^{n+1}-p^0</math><li><math>=p^{n+1}-1</math><li><math>\Leftrightarrow</math><li><math>p^0+p^1+p^2+...+p^n = \frac{p^{n+1}-1}{p-1}</math></ul>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>):<br>

<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 (mathematics)|limit]] <math>\frac{1}{1-p}</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 (mathematics)|limit]] <math>\frac{1}{1-p}</math>.

 

[[Category:mathematics]]

[[Category:mathematics]]