To find the common ratio ($r$) for a geometric sequence, divide any term by the preceding term. Let’s compute the ratio step by step for the given sequence:

The sequence is: $-6, -\frac{1}{6}, -\frac{1}{216}, -\frac{1}{7,776}, -\frac{1}{279,936}.$

Step 1: Calculate the ratio $r$

1. First ratio ($r_1$): $r_1 = \frac{-\frac{1}{6}}{-6} = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}.$

2. Second ratio ($r_2$): $r_2 = \frac{-\frac{1}{216}}{-\frac{1}{6}} = \frac{\frac{1}{216}}{\frac{1}{6}} = \frac{1}{216} \times 6 = \frac{1}{36}.$

3. Third ratio ($r_3$): $r_3 = \frac{-\frac{1}{7,776}}{-\frac{1}{216}} = \frac{\frac{1}{7,776}}{\frac{1}{216}} = \frac{1}{7,776} \times 216 = \frac{1}{36}.$

4. Fourth ratio ($r_4$): $r_4 = \frac{-\frac{1}{279,936}}{-\frac{1}{7,776}} = \frac{\frac{1}{279,936}}{\frac{1}{7,776}} = \frac{1}{279,936} \times 7,776 = \frac{1}{36}.$

Conclusion

The common ratio for the sequence is: $r = \frac{1}{36}.$

Would you like a detailed explanation on how geometric sequences work?

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Related Questions:

1. What is the formula for the $n$th term of a geometric sequence?
2. How do you determine whether a sequence is geometric?
3. What happens when the common ratio is negative in a geometric sequence?
4. Can the terms in a geometric sequence approach zero?
5. How do you sum an infinite geometric series?

Tip: Always check multiple ratios to confirm consistency when verifying a geometric sequence.