Question: What is the common difference in the following arithmetic sequence? \(\frac{2}{3}, \frac{1}{6},…

What is the common difference in the following arithmetic sequence?

\(\frac{2}{3}, \frac{1}{6}, \frac{1}{3}, \frac{5}{6}, \ldots\)

4

\(\frac{1}{9}\)

\(\frac{1}{2}\)

\(-4\)

Solution

To find the common difference in the arithmetic sequence, subtract one term from the succeeding term. Starting with the first two terms: \[ \frac{1}{6} - \frac{2}{3} \] Convert \(\frac{2}{3}\) to have a common denominator of 6: \[ \frac{2}{3} = \frac{4}{6} \] Subtract the terms: \[ \frac{1}{6} - \frac{4}{6} = -\frac{3}{6} = -\frac{1}{2} \] The common difference is \(-\frac{1}{2}\).