To find the common difference of the arithmetic sequence [tex]\( \frac{1}{6}, \frac{1}{4}, \frac{1}{3}, \ldots \)[/tex], we need to follow these steps:
1. Identify the terms of the sequence:
- First term, [tex]\( a_1 = \frac{1}{6} \)[/tex]
- Second term, [tex]\( a_2 = \frac{1}{4} \)[/tex]
- Third term, [tex]\( a_3 = \frac{1}{3} \)[/tex]
2. Calculate the differences between consecutive terms:
- The difference between [tex]\( a_2 \)[/tex] and [tex]\( a_1 \)[/tex]:
[tex]\[
a_2 - a_1 = \frac{1}{4} - \frac{1}{6}
\][/tex]
To perform this subtraction, find a common denominator for [tex]\( \frac{1}{4} \)[/tex] and [tex]\( \frac{1}{6} \)[/tex]:
- Common denominator of 4 and 6 is 12.
- Convert [tex]\( \frac{1}{4} \)[/tex] and [tex]\( \frac{1}{6} \)[/tex] to have a denominator of 12:
[tex]\[
\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}
\][/tex]
- Subtract the fractions:
[tex]\[
\frac{3}{12} - \frac{2}{12} = \frac{1}{12}
\][/tex]
- The difference between [tex]\( a_3 \)[/tex] and [tex]\( a_2 \)[/tex]:
[tex]\[
a_3 - a_2 = \frac{1}{3} - \frac{1}{4}
\][/tex]
Again, find a common denominator for [tex]\( \frac{1}{3} \)[/tex] and [tex]\( \frac{1}{4} \)[/tex]:
- Common denominator of 3 and 4 is 12.
- Convert [tex]\( \frac{1}{3} \)[/tex] and [tex]\( \frac{1}{4} \)[/tex] to have a denominator of 12:
[tex]\[
\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}
\][/tex]
- Subtract the fractions:
[tex]\[
\frac{4}{12} - \frac{3}{12} = \frac{1}{12}
\][/tex]
3. Verify the common difference:
- The difference between the second term and the first term is [tex]\( \frac{1}{12} \)[/tex].
- The difference between the third term and the second term is also [tex]\( \frac{1}{12} \)[/tex].
Since both differences are equal, we conclude that the sequence has a common difference of:
[tex]\[
\boxed{\frac{1}{12}}
\][/tex]
