Find the common ratio for the geometric sequence.
$6,-3\sqrt{2},3,-\frac{3\sqrt{2}}{2},\dots$

• A. $-\sqrt{2}$
• B. $-\frac{\sqrt{2}}{4}$
• C. $-2\sqrt{2}$
• D. $-\frac{\sqrt{2}}{2}$

Answer: (D)

Verified step by step guidance

1. Recall that the common ratio \( r \) in a geometric sequence is found by dividing any term by the previous term. That is, \( r = \frac{a_{n+1}}{a_n} \).
2. Identify the first two terms of the sequence: \( a_1 = 6 \) and \( a_2 = -3\sqrt{2} \).
3. Calculate the ratio \( r \) by dividing the second term by the first term: \( r = \frac{-3\sqrt{2}}{6} \).
4. Simplify the fraction \( \frac{-3\sqrt{2}}{6} \) by dividing numerator and denominator by 3, resulting in \( r = \frac{-\sqrt{2}}{2} \).
5. Optionally, verify the common ratio by dividing the third term by the second term: \( \frac{3}{-3\sqrt{2}} \), and check if it simplifies to the same \( r \).