To determine the common difference in the given table of values, we need to check the difference between consecutive [tex]\( y \)[/tex]-values. Let's go through this step-by-step.
1. List the [tex]\( y \)[/tex]-values from the table:
[tex]\[
y_1 = 8, \quad y_2 = 23, \quad y_3 = 38, \quad y_4 = 53, \quad y_5 = 68
\][/tex]
2. Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[
y_2 - y_1 = 23 - 8 = 15
\][/tex]
[tex]\[
y_3 - y_2 = 38 - 23 = 15
\][/tex]
[tex]\[
y_4 - y_3 = 53 - 38 = 15
\][/tex]
[tex]\[
y_5 - y_4 = 68 - 53 = 15
\][/tex]
3. Verify if the differences are consistent:
Each difference calculated above is [tex]\( 15 \)[/tex]. Since all the differences between consecutive [tex]\( y \)[/tex]-values are the same, the sequence is arithmetic and the common difference is [tex]\( 15 \)[/tex].
Therefore, the common difference of the associated arithmetic sequence is [tex]\( \boxed{15} \)[/tex].
