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

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.