Question

The explicit formula for the geometric sequence Negative one-ninth, one-third, negative 1, 3, negative 9, ellipsis is f (x) = negative one-ninth (negative 3) Superscript x minus 1. What is the common ratio and recursive formula for this sequence?

Answer 1

Answer:

Common ratio, r=-3

Recursive Formula

$f(n)=-3f(n-1), n\geq 2\\f(1)=-\dfrac{1}{9},$

Step-by-step explanation:

The formula for the geometric sequence: $-\dfrac{1}{9} ,\dfrac{1}{3}, -1,3,-9,\cdots$ is given as:

$f(x)=-\dfrac{1}{9}(-3)^{x-1}$

Common Ratio

Dividing the next terms by the previous terms, we obtain:

$\dfrac{1}{3} \div -\dfrac{1}{9} = \dfrac{1}{3} \times -9 =-3\\3 \div -1 =-3\\-9 \div 3 =-3$

Therefore, the common ratio of the sequence, r=-3

Recursive Formula

We observe that the next term, $f(n)$ is obtained by the multiplication of the previous term. f(n-1) by -3.

Therefore, a recursive formula for the sequence is:

$f(n)=-3f(n-1), n\geq 2\\f(1)=-\dfrac{1}{9},$