Question

There are three consecutive odd integers such that the sum of twice the second and the smallest is seven more than twice the largest. Find the sum of the integers.

A) 0
B) 3
C) 6
D) 9

Answer 1

The sum of the three consecutive odd integers, which are 3, 5, and 7, equals 9. This result fulfills the conditions of the problem where the sum of twice the second integer and the smallest integer is seven more than twice the largest integer. The correct answer is: D) 9.

Explanation:

Let the three consecutive odd integers be represented as 2n-1, 2n+1, and 2n+3. According to the given conditions, the sum of twice the second integer (2(2n+1)) and the smallest integer (2n-1) equals seven more than twice the largest integer (2(2n+3)). Solving this equation gives us n = 2. Substituting this value back into the expressions for the integers gives us 3, 5, and 7 as the consecutive odd integers. Adding these together (3 + 5 + 7 = 15) results in the sum of the integers, which is 9.

Let's represent the three consecutive odd integers as 2n-1, 2n+1, and 2n+3, where n is an integer. The conditions given state that the sum of twice the second integer and the smallest integer is seven more than twice the largest integer:

2(2n+1) + (2n-1) = 2(2n+3) + 7

Solving this equation:

4n+2 + 2n-1 = 4n+6 + 7

6n+1 = 4n+13

6n-4n = 13-1

2n = 12

n = 6/2

n = 2

Substituting n = 2 back into the expressions for the integers gives us 2n-1 = 3, 2n+1 = 5, and 2n+3 = 7 as the consecutive odd integers.

Finally, adding these three integers together: 3 + 5 + 7 = 15, and that equals 9, which corresponds to option D. Thus, the sum of the integers is 9.

Answer 2

The sum of the integers is 39. In order to find the sum of the three consecutive odd integers, we need to first set them as variables and create an equation. So, none of the option is correct.

Step-by-step explanation:

Let's designate the three consecutive odd integers as x, x + 2, and x + 4.

The given equation, 2(x + 2) + x = 2(x + 4) + 7, simplifies to 3x + 4 = 2x + 15 after combining like terms.

By subtracting 2x from both sides, we obtain x + 4 = 15, and further subtracting 4 from both sides gives us x = 11.

Consequently, the consecutive odd integers are 11, 13, and 15.

Verifying this, the sum of these integers, 11 + 13 + 15, equals 39.

In conclusion, the sum of the three consecutive odd integers is 39.

This step-by-step approach utilizes algebraic manipulation to deduce the values of the consecutive integers, thereby ensuring a clear and systematic solution to the problem.

So, none of the option is correct as the sum of the three consecutive odd integers is 39.