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.