In a geometric sequence, the common ratio can be found by dividing any term by the preceding term.
Let's find the common ratio using the first two terms of the given sequence:
- First term (a_1): 2
- Second term (a_2): \( \frac{1}{2} \)
The common ratio \( r \) can be calculated as follows:
\[ r = \frac{a_2}{a_1} = \frac{\frac{1}{2}}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]
Now, let's verify this by calculating the ratio between another pair of terms.
Using the third and second terms:
- Second term (a_2): \( \frac{1}{2} \)
- Third term (a_3): \( \frac{1}{8} \)
\[ r = \frac{a_3}{a_2} = \frac{\frac{1}{8}}{\frac{1}{2}} = \frac{1}{8} \times \frac{2}{1} = \frac{2}{8} = \frac{1}{4} \]
Using the fourth and third terms:
- Third term (a_3): \( \frac{1}{8} \)
- Fourth term (a_4): \( \frac{1}{32} \)
\[ r = \frac{a_4}{a_3} = \frac{\frac{1}{32}}{\frac{1}{8}} = \frac{1}{32} \times \frac{8}{1} = \frac{8}{32} = \frac{1}{4} \]
In each case, we find that the common ratio \( r \) is consistently \( \frac{1}{4} \).
Therefore, the common ratio of the sequence is:
Start Fraction 1 over 4 End Fraction \( \frac{1}{4} \)