Definition of Geometric Sequences
A geometric progression, also known as a geometric sequence, is an ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio
Thus, the general form of a geometric sequence is:
The
Such a geometric sequence also follows the recursive relation:
for every integer
Behavior of Geometric Sequences
Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a geometric series may be negative, resulting in an alternating sequence. An alternating sequence will have numbers that switch back and forth between positive and negative signs. For instance:
The behavior of a geometric sequence depends on the value of the common ratio. If the common ratio is:
- Positive, the terms will all be the same sign as the initial term
- Negative, the terms will alternate between positive and negative
- Greater than
$1$ , there will be exponential growth towards positive infinity ($+\infty$ ) $1$ , the progression will be a constant sequence- Between
$-1$ and$1$ but not$0$ , there will be exponential decay toward$0$ $-1$ , the progression is an alternating sequence (see alternating series)- Less than
$-1$ , for the absolute values there is exponential growth toward positive and negative infinity (due to the alternating sign)
Geometric sequences (with common ratio not equal to
An interesting result of the definition of a geometric progression is that for any value of the common ratio, any three consecutive terms