If the mean of `x+2,\ 2x+3,\ 3x+4,\ 4x+5\ i s\ x+2,` find `x`

Options

A. 0
B. 1
C. -1
D. 2

Explanation

To solve the problem, we need to find the value of \( x \) given that the mean of the observations \( x + 2, 2x + 3, 3x + 4, 4x + 5 \) is equal to \( x + 2 \).

### Step-by-Step Solution:

1. **Identify the Observations**:
The observations given are:
\[ x + 2, \quad 2x + 3, \quad 3x + 4, \quad 4x + 5 \]

2. **Count the Number of Observations**:
There are 4 observations in total.

3. **Calculate the Sum of the Observations**:
We need to find the sum of the observations:
\[ (x + 2) + (2x + 3) + (3x + 4) + (4x + 5) \]
Simplifying this:
\[ = x + 2 + 2x + 3 + 3x + 4 + 4x + 5 \]
Combine like terms:
\[ = (x + 2x + 3x + 4x) + (2 + 3 + 4 + 5) = 10x + 14 \]

4. **Set Up the Mean Equation**:
The mean is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} \]
Substituting the values we have:
\[ \frac{10x + 14}{4} = x + 2 \]

5. **Cross Multiply to Eliminate the Fraction**:
Cross multiplying gives us:
\[ 10x + 14 = 4(x + 2) \]

6. **Expand the Right Side**:
Expanding the right side:
\[ 10x + 14 = 4x + 8 \]

7. **Rearrange the Equation**:
Move all terms involving \( x \) to one side and constant terms to the other side:
\[ 10x - 4x = 8 - 14 \]
This simplifies to:
\[ 6x = -6 \]

8. **Solve for \( x \)**:
Divide both sides by 6:
\[ x = -1 \]

### Final Answer:
The value of \( x \) is \( -1 \).

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