Question

What is the recursive rule for the sequence 2, -1, 1/2 , -1/4

Answer 1

`a_n=(-1)/(2)a_(n-1)`

`a_1=2`

Explanation:

The recursive rule is a term defined in terms of other terms in the sequence.

The is a geometric sequence because it has a common ratio.

The common ratio can be found by dividing a term by previous term.

For example, all of these are equal:

`(-1)/(2)`

`((1)/(2))/(-1)`

`((-1)/(4))/((1)/(2))`

They are all equal to
`(-1)/(2)`.

So we are saying:

`\frac{\text{term}}{\text{previous term}}}=(-1)/(2)`

More formally:

`(a_n)/(a_(n-1))=(-1)/(2)`.

Multiply both sides by
`a_(n-1)`:

`a_n=(-1)/(2)a_(n-1)`

When doing recursive form, you need to state a term of the sequence (or more depending on the recursive form you have).

So the first term is 2.

So the full thing for the answer is:

`a_n=(-1)/(2)a_(n-1)`

`a_1=2`