To determine the common difference in an arithmetic sequence, we need to confirm that the sequence actually is arithmetic. This can be done by checking that the difference between consecutive terms is constant.
Given the sequence:
[tex]\[
\frac{2}{3}, \frac{1}{6}, -\frac{1}{3}, -\frac{5}{6}, \ldots
\][/tex]
1. Calculate the difference between the second term and the first term:
[tex]\[
\frac{1}{6} - \frac{2}{3}
\][/tex]
Convert [tex]\(\frac{2}{3}\)[/tex] to a fraction with a common denominator:
[tex]\[
\frac{2}{3} = \frac{4}{6}
\][/tex]
Subtract the fractions:
[tex]\[
\frac{1}{6} - \frac{4}{6} = \frac{1 - 4}{6} = \frac{-3}{6} = -\frac{1}{2}
\][/tex]
2. Calculate the difference between the third term and the second term:
[tex]\[
-\frac{1}{3} - \frac{1}{6}
\][/tex]
Convert [tex]\(-\frac{1}{3}\)[/tex] to a fraction with a common denominator:
[tex]\[
-\frac{1}{3} = -\frac{2}{6}
\][/tex]
Subtract the fractions:
[tex]\[
-\frac{2}{6} - \frac{1}{6} = \frac{-2 - 1}{6} = \frac{-3}{6} = -\frac{1}{2}
\][/tex]
3. Calculate the difference between the fourth term and the third term:
[tex]\[
-\frac{5}{6} - (-\frac{1}{3})
\][/tex]
Convert [tex]\(-\frac{1}{3}\)[/tex] to a fraction with a common denominator:
[tex]\[
-\frac{1}{3} = -\frac{2}{6}
\][/tex]
Subtract the fractions:
[tex]\[
-\frac{5}{6} - (-\frac{2}{6}) = -\frac{5}{6} + \frac{2}{6} = \frac{-5 + 2}{6} = \frac{-3}{6} = -\frac{1}{2}
\][/tex]
Since the difference between each consecutive pair of terms is consistently [tex]\(-\frac{1}{2}\)[/tex], this confirms that the sequence is arithmetic with a common difference of [tex]\(-\frac{1}{2}\)[/tex].
Thus, the common difference in the arithmetic sequence [tex]\(\frac{2}{3}, \frac{1}{6}, -\frac{1}{3}, -\frac{5}{6}, \ldots\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex].
The correct answer is:
[tex]\[
-\frac{1}{2}
\][/tex]
