Let's break this down step by step:

Items 6 – 10: Finding the Specified Term of a Geometric Sequence

For each geometric sequence, the nth term can be found using the formula:

$a_n = a_1 \times r^{(n-1)}$

Where:

• $a_n$ is the nth term.
• $a_1$ is the first term.
• $r$ is the common ratio.
• $n$ is the term number.

Let's solve each problem.

6. Sequence: 3, 6, 12, … Find $a_7$

• $a_1 = 3$
• $r = \frac{6}{3} = 2$
• $n = 7$

$a_7 = 3 \times 2^{(7-1)} = 3 \times 2^6 = 3 \times 64 = 192$

Answer: $a_7 = 192$

7. Sequence: 10, 5, $\frac{5}{2}$, … Find $a_6$

• $a_1 = 10$
• $r = \frac{5}{10} = \frac{1}{2}$
• $n = 6$

$a_6 = 10 \times \left(\frac{1}{2}\right)^{(6-1)} = 10 \times \left(\frac{1}{2}\right)^5 = 10 \times \frac{1}{32} = \frac{10}{32} = \frac{5}{16}$

Answer: $a_6 = \frac{5}{16}$

8. Sequence: 1, 3, 9, … Find $a_9$

• $a_1 = 1$
• $r = \frac{3}{1} = 3$
• $n = 9$

$a_9 = 1 \times 3^{(9-1)} = 3^8 = 6561$

Answer: $a_9 = 6561$

9. Sequence: 2, –1, $\frac{1}{2}$, … Find $a_{10}$

• $a_1 = 2$
• $r = \frac{-1}{2} = -\frac{1}{2}$
• $n = 10$

$a_{10} = 2 \times \left(-\frac{1}{2}\right)^{(10-1)} = 2 \times \left(-\frac{1}{2}\right)^9 = 2 \times -\frac{1}{512} = -\frac{2}{512} = -\frac{1}{256}$

Answer: $a_{10} = -\frac{1}{256}$

10. Sequence: 100, 20, 4, … Find $a_8$

• $a_1 = 100$
• $r = \frac{20}{100} = \frac{1}{5}$
• $n = 8$

$a_8 = 100 \times \left(\frac{1}{5}\right)^{(8-1)} = 100 \times \left(\frac{1}{5}\right)^7 = 100 \times \frac{1}{78125} = \frac{100}{78125} = \frac{4}{3125}$

Answer: $a_8 = \frac{4}{3125}$

Items 11 – 15: Inserting Geometric Means

To insert geometric means, find the common ratio $r$ by dividing the last term by the first and taking the appropriate root depending on the number of means to be inserted. Then multiply each preceding term by $r$ to find the next term.

11. Sequence: 3, ____, 8

• We need to find $r$ such that $8 = 3r^2$.

$r^2 = \frac{8}{3}, \quad r = \sqrt{\frac{8}{3}} = \frac{2\sqrt{6}}{3}$

The geometric mean is $3 \times \frac{2\sqrt{6}}{3} = 2\sqrt{6}$.

**Answer: ( 3, , 2\sqrt{6}, , 8