QUESTION 3 $$ C: 1,2,4,8,16,32, \ldots $$ PART A Use the first 5 terms of each sequence to state if the sequence is arithmetic, geometric, or neither. A Arithmetic https:/lapp.kiddom.co/curriculum/IM360M.KY.A2/node/21d09885-ea04-4277-948f-51e79f925731:782dea77-a6c2-11ef-8a89-06dd2b7bf731:b6068ed1-... $$ 2 / 5 $$ 1/14/25, 7:08 PM | Kiddom B Geometric C Neither PART B Next, define the sequence recursively using function notation. $$ \qquad $$ $$ \qquad $$ $$ \qquad $$ $$ \qquad $$

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**QUESTION 3**

$$
C: 1, 2, 4, 8, 16, 32, \ldots
$$

---

### PART A

**Determine if the sequence is arithmetic, geometric, or neither using the first 5 terms.**

**Given Sequence:** 1, 2, 4, 8, 16

1. **Arithmetic Sequence:**
   - **Definition:** A sequence is arithmetic if the difference between consecutive terms is constant.
   - **Check:**
     - $$2 - 1 = 1$$
     - $$4 - 2 = 2$$
     - $$8 - 4 = 4$$
     - $$16 - 8 = 8$$
   - **Observation:** The differences (1, 2, 4, 8) are not constant.
   - **Conclusion:** *Not an arithmetic sequence.*

2. **Geometric Sequence:**
   - **Definition:** A sequence is geometric if the ratio between consecutive terms is constant.
   - **Check:**
     - $$\frac{2}{1} = 2$$
     - $$\frac{4}{2} = 2$$
     - $$\frac{8}{4} = 2$$
     - $$\frac{16}{8} = 2$$
   - **Observation:** The ratio is consistently 2.
   - **Conclusion:** *Geometric sequence with a common ratio of 2.*

3. **Neither:**
   - Since it's identified as geometric, this option is not applicable.

**Final Answer for Part A:**  
**B. Geometric**

---

### PART B

**Define the sequence recursively using function notation.**

A **recursive definition** specifies the first term (or terms) and a rule to obtain subsequent terms from the preceding ones.

**For the given geometric sequence:**

1. **Base Case:**
   - $$ f(1) = 1 $$  
     *(The first term of the sequence is 1.)*

2. **Recursive Step:**
   - $$ f(n) = 2 \cdot f(n-1) $$ for $$ n > 1 $$  
     *(Each term is twice the previous term.)*

**Complete Recursive Definition:**

$$
f(1) = 1
$$
$$
f(n) = 2 \times f(n-1) \quad \text{for } n > 1
$$

**Alternatively, using $$ a_n $$ notation:**

$$
a_1 = 1
$$
$$
a_n = 2a_{n-1} \quad \text{for } n > 1
$$

**Summary for Part B:**  
Define the sequence recursively by setting the first term to 1 and each subsequent term to twice the previous term. In function notation:

$$
f(1) = 1
$$
$$
f(n) = 2\,f(n-1) \quad \text{for } n > 1
$$
**PART A:** The sequence is **geometric**. **PART B:** The sequence can be defined recursively as: $$ f(1) = 1 $$ $$ f(n) = 2 \times f(n-1) \quad \text{for } n > 1 $$

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