Question: Repeat the following procedure for the four given numbers. Multiply the number by $12$. Add $8$ to the product. Divide this sum by $2$. Subtract $4$ from the quotient.
Part 1 of 9
The 1st number is $2$. The result is $12$.
Part 2 of 9
The 2nd number is $4$. The result is $24$.
Part 3 of 9
The 3rd number is $8$. The result is $48$.
Part 4 of 9
The 4th number is $12$. The result is $72$.
Part 5 of 9
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 $6 n$. (Simplify your answer.)
Part 6 of 9
b. Represent the original number as $n$, and use deductive reasoning to prove the conjecture in part (a). Multiply the number by $12$. $n$ plus $9$
Part 7 of 9
Add $8$ to the product. enter your response here
Part 8 of 9
Divide the sum by $2$. enter your response here (Simplify your answer.)
Part 9 of 9
Subtract $4$ from the quotient. enter your response here
1. **State the problem:** We are given a procedure to apply to numbers: multiply by $12$, add $8$, divide by $2$, then subtract $4$. We want to verify the results for given numbers and find a general formula.
2. **Apply the procedure to each number:**
- For $n=2$:
$$12 \times 2 = 24$$
$$24 + 8 = 32$$
$$\frac{32}{2} = 16$$
$$16 - 4 = 12$$
Result: $12$
- For $n=4$:
$$12 \times 4 = 48$$
$$48 + 8 = 56$$
$$\frac{56}{2} = 28$$
$$28 - 4 = 24$$
Result: $24$
- For $n=8$:
$$12 \times 8 = 96$$
$$96 + 8 = 104$$
$$\frac{104}{2} = 52$$
$$52 - 4 = 48$$
Result: $48$
- For $n=12$:
$$12 \times 12 = 144$$
$$144 + 8 = 152$$
$$\frac{152}{2} = 76$$
$$76 - 4 = 72$$
Result: $72$
3. **Conjecture:** The result seems to be $6n$.
4. **Proof using algebra:**
Start with original number $n$.
- Multiply by $12$:
$$12n$$
- Add $8$:
$$12n + 8$$
- Divide by $2$:
$$\frac{12n + 8}{2} = \frac{12n}{2} + \frac{8}{2} = 6n + 4$$
- Subtract $4$:
$$6n + 4 - 4 = 6n$$
5. **Conclusion:** The result of the procedure for any number $n$ is $6n$.
**Final answers for parts 7, 8, 9:**
7. Add $8$ to the product:
$$12n + 8$$
8. Divide the sum by $2$:
$$\frac{12n + 8}{2} = 6n + 4$$
9. Subtract $4$ from the quotient:
$$6n + 4 - 4 = 6n$$