`6,-2,2/3,-2/9,...` Write the next two apparent terms of the sequence. Describe the patterns used to find these terms.

Expert Answers
lemjay eNotes educator| Certified Educator

`6, -2, 2/3, -2/9`

To determine the next two terms, identify if it is an arithmetic or geometric sequence.

Take note that an arithmetic sequence has a common difference. While a geometric sequence have a common ratio.

To find the common difference, subtract the successive terms.

`-2-6=-8`

`2/3-(-2)=8/3`

`-2/9-2/3=-8/9`

Since the three pairs of consecutive terms do not have the same result, the given sequence is not an arithmetic sequence.

To find the common ratio, divide the consecutive terms.

`-2/6=-1/3`

`(2/3)/(-2) = -1/3`

`(-2/9)/(2/3)=-1/3`

Since the result are the same, the given sequence is geometric. Its common ratio is `-1/3` .

So the 5th term of the geometric sequence is:

`-2/9*(-1/3) = 2/27`

And its 6th term is:

`2/27*(-1/3)=-2/81`

Therefore, the next two terms of the given sequence are `2/27` and `-2/81`.