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The sequence provided is \( 14, 7, \frac{7}{2}, \dots \). We need to find the 6th term.

This appears to be a geometric sequence where each term is obtained by multiplying the previous one by a constant ratio. Let's calculate the common ratio \( r \).

1. The second term divided by the first term gives us:
   \[
   r = \frac{7}{14} = \frac{1}{2}
   \]

2. To confirm, divide the third term by the second term:
   \[
   r = \frac{\frac{7}{2}}{7} = \frac{1}{2}
   \]
   
The common ratio is \( \frac{1}{2} \). Now, we use the formula for the \(n\)-th term of a geometric sequence:
\[
a_n = a_1 \cdot r^{n-1}
\]
where:
- \( a_1 = 14 \)
- \( r = \frac{1}{2} \)
- \( n = 6 \)

Substitute the values:
\[
a_6 = 14 \cdot \left( \frac{1}{2} \right)^{5} = 14 \cdot \frac{1}{32} = \frac{14}{32} = 0.4375
\]

Thus, the 6th term is \( 0.4375 \).

Do you need more details or have any questions?

Here are 5 related questions to consider:
1. What would be the 10th term in this sequence?
2. How can you verify if a sequence is geometric?
3. What is the sum of the first 6 terms of this sequence?
4. How would the 6th term change if the ratio were \( \frac{1}{3} \) instead of \( \frac{1}{2} \)?
5. How can we generalize the formula for an arithmetic sequence?

**Tip:** Always verify the common ratio by checking multiple terms, not just the first two.

The first three terms of a sequence are given: 14, 7, 7/2, and we need to find the 6th term.

Learn how to calculate the 6th term of the geometric sequence 14, 7, 7/2 using the common ratio and the geometric sequence formula. Suitable for high school students, this step-by-step guide explores key mathematical concepts, including sequences and common ratios.

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