Question

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

What is the explicit rule for this sequence?



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an=

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({ (3)/(4) })^(n - 1)`


Step-by-step explanation


The recursive rule for the sequence is given as,



`a_n= (3)/(4) a_(n-1)`


Where,


`image`


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= (3)/(4) a_(2-1)`



This implies that,


`a_2= (3)/(4) a_(1)`


This will give us



`a_2= (3)/(4) (8)`




`a_2= (3)/(1) (2)`





`a_2= 3 * 2`



`a_2= 6`


The common ratio is



`r = (a_2)/(a_1)`


`r = (6)/(8)`
This reduces to



`r = (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({ (3)/(4) })^(n - 1)`