What sequence of numbers follows the form: a_n=ar^(n-1) in mathematics?

The sequence of numbers described by the formula ( a_n = ar^{(n-1)} ) represents a geometric progression. In a geometric progression, each term is derived by multiplying the previous term by a constant factor, known as the common ratio, denoted here by ( r ). The first term is represented by ( a ), and as ( n ) increases, the value of each subsequent term corresponds to the product of the first term and the common ratio raised to the power of ( (n-1) ).

This relationship highlights the characteristic nature of geometric sequences, where growth or decline occurs exponentially relative to the common ratio. For instance, if the common ratio is greater than one, the sequence increases, while if it is between zero and one, the sequence decreases.

Understanding this rule is crucial in various mathematical applications, such as finance, population studies, and many areas of engineering, where quantities grow or diminish by a consistent proportional rate. This contrasts with other types of sequences, which do not operate under this multiplicative principle.