Answer: The required common ratio for the given geometric sequence is [tex]\dfrac{1}{5}.[/tex]
Step-by-step explanation: We are given to find the common ratio for the following geometric sequence :
225, 45, 9, . . .
We know that
in a geometric sequence, the ratio of any term with the preceding term is the common ratio of the sequence.
For the given geometric sequence, we have
a(1) = 225, a(2) = 45, a(3) = 9, etc.
So, the common ratio (r) is given by
[tex]r=\dfrac{a(2)}{a(1)}=\dfrac{a(3)}{a(2)}=~~.~~.~~.~~.[/tex]
We have
[tex]\dfrac{a(2)}{a(1)}=\dfrac{45}{225}=\dfrac{1}{5},\\\\\\\dfrac{a(3)}{a(2)}=\dfrac{9}{45}=\dfrac{1}{5},~etc.[/tex]
Therefore, we get
[tex]r=\dfrac{1}{5}.[/tex]
Thus, the required common ratio for the given geometric sequence is [tex]\dfrac{1}{5}.[/tex]
