Find three consecutive integers such that the sum of the three is equal to 2 times the sum of the first two 
integers

Let the three consecutive integers be \( x \), \( x+1 \), and \( x+2 \).

### Step 1: Set up the equation
The sum of the three integers is:

\[
x + (x+1) + (x+2)
\]

This simplifies to:

\[
3x + 3
\]

Next, the sum of the first two integers is:

\[
x + (x+1) = 2x + 1
\]

According to the problem, the sum of the three integers is equal to 2 times the sum of the first two integers. Therefore, we set up the following equation:

\[
3x + 3 = 2(2x + 1)
\]

### Step 2: Solve the equation
Expand and simplify the equation:

\[
3x + 3 = 4x + 2
\]

Now, subtract \(3x\) from both sides:

\[
3 = x + 2
\]

Subtract 2 from both sides:

\[
x = 1
\]

### Step 3: Find the three integers
If \( x = 1 \), the three consecutive integers are:

\[
1, 2, 3
\]

### Step 4: Verify the solution
The sum of these three integers is:

\[
1 + 2 + 3 = 6
\]

The sum of the first two integers is:

\[
1 + 2 = 3
\]

Two times the sum of the first two integers is:

\[
2 \times 3 = 6
\]

Since the sum of the three integers is indeed equal to 2 times the sum of the first two integers, the solution is correct.

Thus, the three consecutive integers are \( 1 \), \( 2 \), and \( 3 \).

Would you like more details, or do you have any questions?

### Related questions:
1. What are the consecutive integers if the sum of three is three times the sum of the first two?
2. How do you solve for four consecutive integers where their sum equals three times the first integer?
3. What is the sum of three consecutive odd integers that equals a given number?
4. How do you express the sum of three consecutive integers algebraically?
5. What if the sum of four consecutive integers is equal to twice the sum of the first two integers?

### Tip: When solving word problems, always define variables clearly and write out the relationships as equations.

Find three consecutive integers such that the sum of the three is equal to 2 times the sum of the first two integers.

Learn how to find three consecutive integers where their sum is equal to twice the sum of the first two integers. This problem involves algebraic concepts like linear equations and properties of consecutive integers, suitable for students in Grades 6-8.

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