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
enter your response here
.
Part 2 of 9
The 2nd number is 4.
The result is
enter your response here
.
Part 3 of 9
The 3rd number is 8.
The result is
enter your response here
.
Part 4 of 9
The 4th number is 12.
The result is
enter your response here
.
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
enter your response here
. (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.
enter your response here
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 four numbers: multiply the number by 12, add 8, divide by 2, then subtract 4. We will find the result for each number and then find a general formula.
2. **Apply the procedure to each number:**
**For the 1st number, 2:**
Multiply by 12: $2 \times 12 = 24$
Add 8: $24 + 8 = 32$
Divide by 2: $\frac{32}{2} = 16$
Subtract 4: $16 - 4 = 12$
**For the 2nd number, 4:**
Multiply by 12: $4 \times 12 = 48$
Add 8: $48 + 8 = 56$
Divide by 2: $\frac{56}{2} = 28$
Subtract 4: $28 - 4 = 24$
**For the 3rd number, 8:**
Multiply by 12: $8 \times 12 = 96$
Add 8: $96 + 8 = 104$
Divide by 2: $\frac{104}{2} = 52$
Subtract 4: $52 - 4 = 48$
**For the 4th number, 12:**
Multiply by 12: $12 \times 12 = 144$
Add 8: $144 + 8 = 152$
Divide by 2: $\frac{152}{2} = 76$
Subtract 4: $76 - 4 = 72$
3. **Write a conjecture:** Let the original number be $n$. The procedure is:
$$
\text{Result} = \left(\frac{12n + 8}{2}\right) - 4
$$
Simplify inside the parentheses:
$$
\frac{12n + 8}{2} = \frac{12n}{2} + \frac{8}{2} = 6n + 4
$$
Subtract 4:
$$
6n + 4 - 4 = 6n
$$
So the conjecture is: **The result is $6n$.**
4. **Prove the conjectation using deductive reasoning:**
- Multiply the number by 12:
$$
12n
$$
- Add 8:
$$
12n + 8
$$
- Divide the sum by 2:
$$
\frac{12n + 8}{2} = 6n + 4
$$
- Subtract 4:
$$
6n + 4 - 4 = 6n
$$
This matches the conjecture.
**Final answers:**
- For $n=2$, result is $12$
- For $n=4$, result is $24$
- For $n=8$, result is $48$
- For $n=12$, result is $72$
- General formula: result = $6n$