ANSWER
[tex]a_n=\frac{4}{3}a_
{n - 1}, a_1=\frac{1}{2}[/tex]
EXPLANATION
The explicit rule for the given geometric sequence is
[tex]a_n = \frac{1}{2} ( \frac{4}{3} ) ^{n - 1} [/tex]
The first term of the geometric sequence can be obtained by substituting n=1.
[tex]a_1= \frac{1}{2}(\frac{4}{3} )^{1 - 1}[/tex]
[tex]a_1= \frac{1}{2} ( \frac{4}{3} )^{0}[/tex]
[tex]a_1= \frac{1}{2} [/tex]
The common ratio is
[tex]\frac{4}{3}[/tex]
To get the subsequent terms we multiply the previous terms by
[tex] \frac{4}{3}[/tex]
The recursive rule is therefore,
[tex]a_n=\frac{4}{3}a_
{n - 1}, a_1=\frac{1}{2}[/tex]
