a. Write a conjecture that relates the result of the process to the original number selected. Represent the original number as .
The result is 4 n . (Simplify your answer.)
b. Represent the original number as n , and use deductive reasoning to prove the conjecture in part (a).
Multiply the number by 8 .

Question

a. Write a conjecture that relates the result of the process to the original number selected. Represent the original number as

.
The result is 4 n . (Simplify your answer.)
b. Represent the original number as n , and use deductive reasoning to prove the conjecture in part (a).
Multiply the number by 8 .

UpStudy AI · Accepted Answer

Certainly! Let's address each part of your question step by step.

---

### **a. Conjecture Relating the Process Result to the Original Number**

**Conjecture:**  
When the original number is represented by $$ n $$, the result of the process is four times the original number. Mathematically, this can be expressed as:
$$
\text{Result} = 4n
$$

*Simplified Answer:*  
$$
R = 4n
$$

---

### **b. Proof of the Conjecture Using Deductive Reasoning**

**Given:**  
- Original number: $$ n $$  
- Process: Multiply the number by 8.

**Objective:**  
Prove that the result of the process is $$ 4n $$.

**Proof:**

1. **Start with the Original Number:**  
   Let the original number be $$ n $$.

2. **Apply the Given Process:**  
   The process involves multiplying the number by 8.
   $$
   \text{Result} = 8 \times n = 8n
   $$

3. **Compare with the Conjecture:**  
   The conjecture states that the result should be $$ 4n $$. However, according to the process:
   $$
   8n \neq 4n \quad \text{(unless $$ n = 0 $$)}
   $$
   This indicates a discrepancy between the conjecture and the process.

4. **Resolving the Discrepancy:**  
   To align the conjecture with the process, we need to adjust our conjecture based on the given process.

- **Revised Conjecture:**  
     If the process is to multiply the number by 8, then the conjecture should reflect that the result is eight times the original number.
     $$
     R = 8n
     $$

5. **Conclusion:**  
   Given the process of multiplying the original number $$ n $$ by 8, the accurate conjecture should be:
   $$
   R = 8n
   $$
   Therefore, the initial conjecture $$ R = 4n $$ does not hold true under the provided process. The correct relationship is that the result is eight times the original number.

---

**Final Note:**  
There appears to be an inconsistency between the initial conjecture $$ R = 4n $$ and the described process of multiplying the number by 8. To maintain logical consistency, the conjecture should be updated to $$ R = 8n $$ based on the process provided.
The result of multiplying the original number $$ n $$ by 8 is $$ 8n $$.

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