Using Recursive Formulas for Geometric Sequences
A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. As with any recursive formula, the initial term must be given.
recursive formula for a geometric sequence
The recursive formula for a geometric sequence with common ratio [latex]r[/latex] and first term [latex]a_1[/latex] is
[latex]a_n = ra_{n-1}, n \ge 2[/latex]
[latex]\left\{6,9,13.5,20.25,\dots\right\}[/latex]
[latex]\\[/latex]
Using the definition of the geometric sequence [latex]\left\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\right\}[/latex], we have [latex]a_n=a_1\left(r\right)^{n-1}[/latex].
[latex]\\[/latex]
Recall the form of an exponential function [latex]f(x)=a\left(b\right)^x[/latex].