Question

What is the recursive formula for the geometric sequence with this explicit formula

an=9*(-1/3)^(n-1)

Answer 1

a_0 = -27

a_n = a_(n-1) * (-1/3)

Explanation:

First evaluate given formula at n=0 and specify that as starting value

Then find how to get from n-1 to n by comparing two values. In this case the next value is formed by multiplying by -1/3.

Answer 2

`a_n = a_(n-1) \cdot (-(1)/(3))`

Explanation:

The explicit formula for the geometric sequence is given by:

`a_n = a_1 \cdot r^(n-1)`

where,

`a_1` is the first term

r is the common ratio to the following terms.

As per the statement:

Given the explicit formula for geometric sequence:

`a_n = 9 \cdot ((-1)/(3))^(n-1)`

On comparing with [1] we have;

`a_1 = 9` and
`r = -(1)/(3)`

The recursive formula for geometric sequence is given by:

`a_n = a_(n-1) \cdot r`

Substitute the given values we have;

`a_n = a_(n-1) \cdot (-(1)/(3))`

Therefore, the recursive formula for the geometric sequence is,
`a_n = a_(n-1) \cdot (-(1)/(3))`