What is the explicit rule for this geometric sequence? a1=4; an=1/3⋅an−1A) an=1/3⋅4^n−1B) an=4(1/3)^n−1C) an=4(1/3)^nD) an=1/3⋅4^n

ANSWER

The correct answer is B)

[tex]a_n=4( \frac{1}{3} )^{n-1}[/tex]


EXPLANATION

The recursive formula for the geometric sequence is

[tex]a_n= \frac{1}{3} a_{n-1}[/tex]
where,

[tex]a_1 = 4[/tex]

This implies that,

[tex]a_2= \frac{1}{3} a_{2-1}[/tex]


[tex]a_2= \frac{1}{3} a_{1}[/tex]
[tex]a_2= \frac{1}{3} \times 4 = \frac{4}{3} [/tex]



The explicit rule is of the form,

[tex]a_n=a_1r^{n-1}[/tex]
where

[tex]r = \frac{ a_{2}}{ a_{1}} [/tex]

[tex]r = \frac{ \frac{4}{3} }{ 4} [/tex]


[tex]r = \frac{4}{3} \times \frac{1}{4} = \frac{1}{3} [/tex]


The explicit rule is given by,

[tex]a_n=4( \frac{1}{3} )^{n-1}[/tex]