Can someone help me with geometric sequences?

1/8, -1/4, 1/2, -1, 2, -4, 8

or

-1/8, 1/4, -1/2, 1, -2, 4, -8

The generic form that I use for the n'th term of a geometric sequence is:

a(n) = ar^(n - 1)

Where a is the first term (1/8)

and r is the common ratio (unknown)

We know that the 7th term is 8 or a(7) = 8, so we can substitute that and solve for the unknown to find the common ratio:

a(n) = (1/8)r^(n - 1)

a(7) = (1/8)r^(7 - 1)

8 = (1/8)r⁶

64 = r⁶

Let's get the square root of both sides to simplify it vs. just trying to get to the 6th root right away:

±8 = r³

Now, this we can solve without external means:

r = ±2

We have common ratios (which is why you have two sets of square to answer).

Now multiply 1/8 by ±2 five times to get the answer:

1/8

1/8 * -2 = -1/4

-1/4 * -2 = 1/2

1/2 * -2 = -1

-1 * -2 = 2

2 * -2 = -4

-4 * -2 = 8

So the values in your blanks are:

-1/4, 1/2, -1, 2, -4

And in the other set:

1/4, 1/2, 1, 2, 4