Calculus of a Single Variable, Chapter 9, 9.1, Section 9.1, Problem 16

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.