Definition §

Geometric Sequence

A Sequence where there is a common ratio between the values, it is a First Order Linear Recurrence Relation, it is not linear, and can’t be steady-state (except when r = 0 technically)

(Explicit) Geometric Sequence Formula §

Where: n is a Absolute Value / first term () = Ratio = Current Term or if

(First-Order Recurrance) Geometric Sequence Formula §

Where: r = Ratio

If this is true, it has a common ratio:

Geometric Sequence - Graphed

left=-100; right=100
---
y=1*1.1^{(x-1)}
y=1*2^{(x-1)}
y=1*1.01^{(x-1)}

Example

Given Ratio: = 0.8, and , both equal 0.8, therefore it is a Geometric Sequence

Querks

Alternating between Positive and Negative §

If the value of is negative, then the sequence will alternate between positive and negative values, for example:

& Dibocle §

This should be used when dealing with as starting value

This should be used when dealing with as starting value

This makes sense by looking at ‘n’ in , if you are finding the n’th term, in other words then in an equation starting at you must take 1 from to make the equation correlate with the value since, originally the power, of would correlate to ( being ) which usually should be the starting value prior to any alterations, however would cause to be multiplied by the r value. This is offset by , causing it to look something like this = , = , = and so on.

However does not require this alteration, since = (beause both zeros are derived from ), = , = and so on. it is okay for and to correlate here since we expect to be altered, since it is not the orign value.