Answer:
Common ratio, r=-3
Recursive Formula
[tex]f(n)=-3f(n-1),$ $n\geq 2\\f(1)=-\dfrac{1}{9},[/tex]
Step-by-step explanation:
The formula for the geometric sequence: [tex]-\dfrac{1}{9} ,\dfrac{1}{3}, -1,3,-9,\cdots[/tex] is given as:
[tex]f(x)=-\dfrac{1}{9}(-3)^{x-1}[/tex]
Common Ratio
Dividing the next terms by the previous terms, we obtain:
[tex]\dfrac{1}{3} \div -\dfrac{1}{9} = \dfrac{1}{3} \times -9 =-3\\3 \div -1 =-3\\-9 \div 3 =-3[/tex]
Therefore, the common ratio of the sequence, r=-3
Recursive Formula
We observe that the next term, [tex]f(n)[/tex] is obtained by the multiplication of the previous term. f(n-1) by -3.
Therefore, a recursive formula for the sequence is:
[tex]f(n)=-3f(n-1),$ $n\geq 2\\f(1)=-\dfrac{1}{9},[/tex]
Answer:
It is in fact B. −3; f(x + 1) = −3(f(x))
Step-by-step explanation:
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