Geometric Sequences Practice 2 worksheet with exercises on identifying, extending, and graphing geometric sequences.

Let's solve each problem step by step from the Geometric Sequences Practice 2 worksheet.

---

1. Is the sequence geometric or not?

Sequence: $ \{-5, 10, -20, \dots\} $

To check if it's geometric, find the common ratio $ r $ between consecutive terms:

- $ \frac{10}{-5} = -2 $
- $ \frac{-20}{10} = -2 $

Since the ratio is constant ($ r = -2 $), this is a geometric sequence.

✔ Answer: Yes, it is geometric.

---

2. Find the next 3 terms in the sequence.

Sequence: $ \{100, 50, 25, \dots\} $

Find the common ratio:
- $ \frac{50}{100} = 0.5 $
- $ \frac{25}{50} = 0.5 $

So $ r = 0.5 $. Multiply each term by 0.5 to get the next:

- $ 25 \times 0.5 = 12.5 $
- $ 12.5 \times 0.5 = 6.25 $
- $ 6.25 \times 0.5 = 3.125 $

✔ Next three terms: $ 12.5, 6.25, 3.125 $

---

5. Write a recursive rule for the nth term of the sequence: $ \{12, 3, 0.75, \dots\} $

First, find the common ratio:
- $ \frac{3}{12} = 0.25 $
- $ \frac{0.75}{3} = 0.25 $

So $ r = 0.25 $

A recursive rule has two parts:
- Initial term: $ a_1 = 12 $
- Recursive formula: $ a_n = a_{n-1} \cdot r $

✔ Recursive rule:
$$
a_1 = 12, \quad a_n = a_{n-1} \cdot 0.25
$$

---

6. Write an explicit rule for the nth term of the sequence: $ \{2, 8, 32, \dots\} $. Then find $ a_{21} $.

Find the common ratio:
- $ \frac{8}{2} = 4 $
- $ \frac{32}{8} = 4 $

So $ r = 4 $, and $ a_1 = 2 $

The explicit formula for a geometric sequence is:
$$
a_n = a_1 \cdot r^{n-1}
$$

So:
$$
a_n = 2 \cdot 4^{n-1}
$$

Now find $ a_{21} $:
$$
a_{21} = 2 \cdot 4^{20}
$$

We can leave it like this or write it as:
$$
a_{21} = 2 \cdot (2^2)^{20} = 2 \cdot 2^{40} = 2^{41}
$$

✔ Explicit rule: $ a_n = 2 \cdot 4^{n-1} $
✔ $ a_{21} = 2 \cdot 4^{20} = 2^{41} $ (approximately $ 2.199 \times 10^{12} $)

---

7. Write an explicit rule for the nth term of the sequence: $ \{6, 18, 54, \dots\} $. Then find $ a_{35} $.

Common ratio:
- $ \frac{18}{6} = 3 $
- $ \frac{54}{18} = 3 $

So $ r = 3 $, $ a_1 = 6 $

Explicit rule:
$$
a_n = 6 \cdot 3^{n-1}
$$

Now $ a_{35} = 6 \cdot 3^{34} $

This is a very large number, but we can leave it in exponential form.

✔ Explicit rule: $ a_n = 6 \cdot 3^{n-1} $
✔ $ a_{35} = 6 \cdot 3^{34} $

---

8. Write a recursive rule for the nth term of the sequence: $ \{225, 45, 9, \dots\} $. Then find $ a_7 $.

Common ratio:
- $ \frac{45}{225} = 0.2 $
- $ \frac{9}{45} = 0.2 $

So $ r = 0.2 = \frac{1}{5} $

Initial term: $ a_1 = 225 $

Recursive rule:
$$
a_1 = 225, \quad a_n = a_{n-1} \cdot 0.2
$$

Now compute up to $ a_7 $:

- $ a_1 = 225 $
- $ a_2 = 225 \cdot 0.2 = 45 $
- $ a_3 = 45 \cdot 0.2 = 9 $
- $ a_4 = 9 \cdot 0.2 = 1.8 $
- $ a_5 = 1.8 \cdot 0.2 = 0.36 $
- $ a_6 = 0.36 \cdot 0.2 = 0.072 $
- $ a_7 = 0.072 \cdot 0.2 = 0.0144 $

✔ Recursive rule: $ a_1 = 225, \quad a_n = a_{n-1} \cdot 0.2 $
✔ $ a_7 = 0.0144 $

---

9. Use the sequence to complete the table. Then graph it on the coordinate plane.

Sequence: $ \{2, 4, 8, 16, 32, \dots\} $

This is geometric with $ a_1 = 2 $, $ r = 2 $

So:
- $ a_n = 2 \cdot 2^{n-1} = 2^n $

Let’s complete the table:

| $ n $ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|--------|---|---|---|---|---|---|---|---|
| $ a(n) $ | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 |

So the next terms are:
- $ a(6) = 64 $
- $ a(7) = 128 $
- $ a(8) = 256 $

✔ Completed table:
| $ n $ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|--------|---|---|---|---|---|---|---|---|
| $ a(n) $ | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 |

#### Graphing:
Plot points: $ (1,2), (2,4), (3,8), (4,16), (5,32), (6,64), (7,128), (8,256) $

You'll notice that the graph curves upward, increasing rapidly — it's an exponential curve.

✔ What do you notice about the graph?
The graph shows exponential growth — it increases rapidly and forms a smooth curved line, not a straight line. This is characteristic of geometric sequences.

---

✔ Final Answers Summary:

1. Yes, it is geometric.
2. Next three terms: 12.5, 6.25, 3.125
5. Recursive rule: $ a_1 = 12, \quad a_n = a_{n-1} \cdot 0.25 $
6. Explicit rule: $ a_n = 2 \cdot 4^{n-1} $; $ a_{21} = 2 \cdot 4^{20} = 2^{41} $
7. Explicit rule: $ a_n = 6 \cdot 3^{n-1} $; $ a_{35} = 6 \cdot 3^{34} $
8. Recursive rule: $ a_1 = 225, \quad a_n = a_{n-1} \cdot 0.2 $; $ a_7 = 0.0144 $
9. Table completed: $ a(6)=64, a(7)=128, a(8)=256 $; Graph shows exponential growth.

Let me know if you'd like help graphing it or drawing it!