Growth factor (math)

The growth factor is the constant quotient of two successive terms of a geometric sequence . The term is mainly used when the episode describes a real exponential growth process. When it comes to the interest on capital or debt, one also speaks of the interest factor . A growth factor of is colloquially referred to as "growth". In financial mathematics , one speaks of the compounding or discounting factor . Growth to mean a growth factor , that is a growth on twice; growth by means of a growth factor , thus a growth of three times, etc. With growth factor is present, however, "negative growth". In financial mathematics then one speaks of discounting - or discounting . The term “growth factor” is not used for geometric sequences with negative . ${\displaystyle q}$ ${\displaystyle q>1}$ ${\displaystyle 100\,\%}$ ${\displaystyle q=2}$ ${\displaystyle 200\,\%}$ ${\displaystyle q=3}$ ${\displaystyle 0<q<1}$ ${\displaystyle q}$

calculation

The growth factor can be calculated from two consecutive terms and a geometric sequence using the following equation: ${\displaystyle a_{n}}$ ${\displaystyle a_{n+1}}$

${\displaystyle q={\frac {a_{n+1}}{a_{n}}}}$

For example, the growth factor of the sequence , , , ... calculated, for example with the members and through . ${\displaystyle a_{0}=5}$ ${\displaystyle a_{1}=15}$ ${\displaystyle a_{2}=45}$ ${\displaystyle a_{3}=135}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$ ${\displaystyle q={\tfrac {a_{2}}{a_{1}}}={\tfrac {45}{15}}=3}$

The following equation can be used to calculate from any two terms and with the distance : ${\displaystyle a_{n}}$ ${\displaystyle a_{n+i}}$ ${\displaystyle i>0}$

${\displaystyle q={\sqrt[{i}]{\frac {a_{n+i}}{a_{n}}}}=\left({\frac {a_{n+i}}{a_{n}}}\right)^{\frac {1}{i}}}$

If and, on the other hand, are members of an error-prone sequence with exponential growth, then this equation is used to determine the geometric mean of the growth factor between members to . Examples: The growth factor of the sequence , , , ... calculated, for example with the members and through . The average growth factor of the faulty sequence , , , is calculated with the members and through . ${\displaystyle a_{n}}$ ${\displaystyle a_{n+i}}$ ${\displaystyle a_{n}}$ ${\displaystyle a_{n+i}}$ ${\displaystyle a_{0}=5}$ ${\displaystyle a_{1}=15}$ ${\displaystyle a_{2}=45}$ ${\displaystyle a_{3}=135}$ ${\displaystyle a_{0}}$ ${\displaystyle a_{3}}$ ${\displaystyle q=\left({\tfrac {a_{3}}{a_{0}}}\right)^{\frac {1}{3}}=\left({\tfrac {135}{5}}\right)^{\frac {1}{3}}=3}$ ${\displaystyle a_{0}=5+1}$ ${\displaystyle a_{1}=15-2}$ ${\displaystyle a_{2}=45+1}$ ${\displaystyle a_{3}=135-1}$ ${\displaystyle a_{0}}$ ${\displaystyle a_{3}}$ ${\displaystyle q=\left({\tfrac {a_{3}}{a_{0}}}\right)^{\frac {1}{3}}=\left({\tfrac {134}{6}}\right)^{\frac {1}{3}}\approx 2,82}$

If the growth rate is known, the growth factor can be calculated with: ${\displaystyle p}$

${\displaystyle q=1+p}$

With the same equation, the growth factor can also be calculated from the percentage growth if you divide its value by 100 beforehand. Example: The growth factor of a geometric sequence with a growth rate of or a growth of is calculated by or . ${\displaystyle p=0{,}1}$ ${\displaystyle 10\,\%}$ ${\displaystyle q=1+0{,}1=1{,}1}$ ${\displaystyle q=1+{\tfrac {10}{100}}=1{,}1}$

Negative growth

For between 0 and 1 there is a "negative growth", i.e. a decrease, because it must be negative. From a financial mathematical point of view , the discounting or discounting factor at the rate of interest is then usually designated with ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle v}$

${\displaystyle p^{*}={\frac {-100p}{100+p}}}$.

Example: With a "negative growth" of is and the interest rate . The discount factor and the discount factor then belong to this interest rate . ${\displaystyle -20\,\%}$ ${\displaystyle q=0{,}8}$ ${\displaystyle p^{*}=25}$ ${\displaystyle p^{*}}$ ${\displaystyle q^{*}=1{,}25}$ ${\displaystyle v^{*}=0{,}8}$

Individual evidence

1. ^ IN Bronštejn, KA Semendjajew, G. Musiol, H. Mühlig: Taschenbuch der Mathematik . 6th edition. Harri Deutsch, Frankfurt am Main 2005, ISBN 978-3-8171-2006-2 .