Question

What is the recursive rule for this geometric sequence?

3, 3/2, 3/4, 3,8,...

choices -
a1= 1/2;an = 3/2 x an - 1
a1= 3/2;an = 1/2 x an -1
a1= 3; an = 1/2 x an -1
a1= 1/2;an = 3x an -1

please explain your answer, if you can

Answer 1

The objective here is to find
`r` (so called common ratio):

`r=a_(n)/a_(n-1)=a_(2)/a_(1)= (3)/(2) : 3 = (3)/(2) * (1)/(3) = (1)/(2)`
So assuming the first element of sequence is 3 (as you mentioned) we now can define the recursive rule for this geometric sequence:

`a_(1)=3; a_(n)=(1)/(2)a_(n-1)`

Answer 2

C.
`a_1=3,a_n=(1)/(2)a_(n-1),n\geq 2`

Explanation:

We are given that

`3,(3)/(2),(3)/(4),(3)/(8)`,..

We have to find the recursive formula rule for this geometric sequence.

`a_1=3`

`a_2=(3)/(2)`

`a_2=3* (1)/(2)=(1)/(2)* a_1`

`a_3=(3)/(4)`

`a_3=(1)/(2)* (3)/(2)`

`a_3=(1)/(2)* a_2`

`a_4=(3)/(8)`

`a_4=(1)/(2)* (3)/(4)`

`a_4=(1)/(2)* a_3`

Therefore, the recursive rule

`a_1=3,a_n=(1)/(2)a_(n-1),n\geq 2`

Option C is true