Question

What is the common difference in the following arithmetic sequence?
\(\frac{3}{4}\), \(\frac{6}{4}\), \(\frac{9}{4}\), . . .

Answer 1

To find the common difference in the arithmetic sequence \(\frac{3}{4}, \frac{6}{4}, \frac{9}{4}, \ldots\), we calculate the difference between the consecutive terms.

The first term is \(\frac{3}{4}\) and the second term is \(\frac{6}{4}\).

Calculating the common difference \(d\):

\[
d = \text{second term} - \text{first term} = \frac{6}{4} - \frac{3}{4}
\]

Now, we will simplify this:

\[
d = \frac{6 - 3}{4} = \frac{3}{4}
\]

Next, we check the difference between the second term \(\frac{6}{4}\) and the third term \(\frac{9}{4}\):

\[
d = \text{third term} - \text{second term} = \frac{9}{4} - \frac{6}{4}
\]

Simplifying this:

\[
d = \frac{9 - 6}{4} = \frac{3}{4}
\]

Since the difference is consistent, we conclude that the common difference in the sequence is

\[
\boxed{\frac{3}{4}}
\]