Question

A recursive rule for a geometric sequence is a1=8;an=34an−1.

Answer 1

The recursive rule is : a1=8; an=34an-1. The explicit rule for this sequence is 8(3/4)^n-1.

Your answer is: 8(3/4)^n-1 

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Answer 2

ANSWER

The explicit rule is,

$a_n=8({ \frac{3}{4} })^{n - 1}$


EXPLANATION


The recursive rule for the sequence is given as,


$a_n= \frac{3}{4} a_{n-1}$


Where,

$a_1=8$


Let us find the next term so that we can use it to find the common ratio.



We put
$n = 2$
into the formula to obtain,


$a_2= \frac{3}{4} a_{2-1}$



This implies that,

$a_2= \frac{3}{4} a_{1}$


This will give us


$a_2= \frac{3}{4} (8)$



$a_2= \frac{3}{1} (2)$




$a_2= 3 \times 2$


$a_2= 6$


The common ratio is


$r = \frac{a_2}{a_1}$

$r = \frac{6}{8}$
This reduces to


$r = \frac{3}{4}$

The explicit rule of the sequence is given by



$a_n=a_1 {r}^{n - 1}$



We substitute the values to obtain,



$a_n=8({ \frac{3}{4} })^{n - 1}$