find k given the geometric sequence: 18,k,2/9

In a geometric sequence, the ratio between consecutive terms is constant. Let the common ratio be \( r \). 

Given the sequence: 
\[
18, k, \frac{2}{9}
\]

1. The common ratio between the first and second terms is:
\[
r = \frac{k}{18}
\]

2. The common ratio between the second and third terms is:
\[
r = \frac{\frac{2}{9}}{k} = \frac{2}{9k}
\]

Since the ratio is constant, we can set these two expressions equal to each other:
\[
\frac{k}{18} = \frac{2}{9k}
\]

Now, solve for \( k \):

1. Cross multiply:
\[
k^2 = 18 \times \frac{2}{9}
\]

2. Simplify the right-hand side:
\[
k^2 = \frac{36}{9} = 4
\]

3. Take the square root of both sides:
\[
k = \pm 2
\]

Thus, \( k \) can be \( 2 \) or \( -2 \).

Would you like further details or have any questions about this solution?

Here are 5 related questions for practice:
1. What is the common ratio \( r \) in this geometric sequence for \( k = 2 \)?
2. If \( k = -2 \), what is the common ratio \( r \)?
3. How do you find the next term in a geometric sequence?
4. What happens to the sequence if the common ratio \( r \) is negative?
5. Can you find \( k \) if the first term was changed to 9?

**Tip:** In a geometric sequence, you can always test your result by checking if the ratio between consecutive terms remains the same throughout the sequence.

Find k given the geometric sequence: 18, k, 2/9

Discover how to solve for k in the geometric sequence 18, k, 2/9 using the concept of common ratios. This detailed explanation involves solving a quadratic equation and understanding geometric sequences. Suitable for students in Grades 8-10.

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