Geometric progression

A geometric sequence, a regular mathematical number sequence with the property that the ratio of two adjacent terms of the sequence is constant.

• 3.1 Compound Interest
• 3.2 equal temperament

Mathematical formulation

The nth term of a geometric sequence with the first member and the quotient is calculated from the formula

Or

So, for example,

Are and be positive real numbers, then each member with the geometric mean of its neighboring links and. This fact is the reason for the term " geometric progression ". The sum of terms of the sequence yields the geometric series. The terms of a geometric sequence can also be from the respective preceding limb calculate serves the following formula:

The first follower, when it is expected to, commonly referred to. When used, however.

Numerical examples

Example 1

The terms of the geometric sequence with the first member and the quotient are

If you just behind the other writes the limbs results

Example 2

The terms of the geometric sequence with the first member and the quotient are

If you just behind the other writes the limbs results

Application Examples

The geometric sequence describes growth processes in which the measured quantity at the time of the measured variable at the time by multiplying results by a constant factor. Examples:

Compound interest

Main article: Compound Interest

At an interest rate of 5 percent of the capital each year are increased by a factor of 1.05. It is a geometric series with the ratio. The number is called interest factor here. With a starting budget of 1000 Euro results

• After one year a capital of

• After two years of a capital

• After three years of a capital

And so on.

Equal temperament

Main article: equal temperament

There are several types, like a musical instrument can be tuned. One is the equal temperament. For her, the frequency ratio between two adjacent notes is always constant. In twelve notes in the octave, the result is here:

Where, for example, the frequency of concert pitch, and the Halbtonschrittentfernung is to pitch. is then the frequency of the required tone semitone with the " Ursprungston ".

The growth factor is then.