Question

How do you find the next three terms in the geometric sequence -16, 4, , , ... ?

Answer 1

Find the common ratio `r` between terms, and multiply by it repeatedly to obtain `-1, 1/4, -1/16` as the next three terms in the sequence.

Explanation:

The general form for a geometric sequence with the first term `a` is `a, ar, ar^2, ar^3, ...` where `r` is a common ratio between terms.

As the first two terms of the geometric sequence given are `-16` and `4`, we have `a = -16` and `ar = 4`.

Then, to find `r`, we simply divide the second term by the first to obtain
`(ar)/a = 4/(-16)`
`=> r = -1/4`

Thus the next three terms in the sequence will be
`ar^2 = 4*(-1/4) = -1`
`ar^3 = -1 * (-1/4) = 1/4`
`ar^4 = 1/4 * (-1/4) = -1/16`

Answer 2

`(x_n)=-16, 4, -1, 1/4, -1/16, ...`

Explanation:

Since it is a geometric sequence `(x_n)`, there is a constant ratio `r=(x_(n+1)/x_n)=4/-16=-1/4`

So if `a=-16` is the first term `x_1`, then general term is given by `x_n=ar^(n-1)=-16*(-1/4)^(n-1)`

Hence the 3rd term, `x_3=-16*(-1/4)^(3-1)=-1`

Similarly `x_4=1/4 and x_5=-1/16`