Geometric Sequences
Rule

Explicit Rule of Geometric Sequences

Every geometric sequence can be described by a function known as the explicit rule, whose input is the position of a term and whose output is the term's value An explicit rule for a geometric sequence has the following general form.

Here, is the first term of the sequence and is the common ratio.

Proof

Proof by Induction
Recall that every geometric sequence has a common ratio Therefore, it is possible to find every term of the sequence by multiplying the first term by this common ratio a particular number of times. Therefore, knowing and is enough to describe the whole geometric sequence.
Interactive applet showing how to rewrite the first five terms of a geometric sequence as an expression involving just the common difference and the first term
It is easier to identify a pattern that can be used to write a general expression for the explicit rule by making a table. Note that by the Zero Exponent Property, is equal to Furthermore, can be written as
Using and

It can be seen that the exponent of the common ratio is always less than the value of the position With this pattern, it is possible to write the explicit rule in the same form as the formula given at the beginning.

Exercises