If the mean of five observations `x,(x+1),(x+2)(x+5)` and `(x+7)` is 11, then the mean of first three observations is

Options

A. 9
B. 11
C. 13
D. None of these

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Write down the observations
The five observations are:
- \( x \)
- \( x + 1 \)
- \( x + 2 \)
- \( x + 5 \)
- \( x + 7 \)

### Step 2: Calculate the mean of these observations
The mean of the observations is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the number of observations is 5, and we know the mean is 11. Therefore, we can set up the equation:
\[ \frac{x + (x + 1) + (x + 2) + (x + 5) + (x + 7)}{5} = 11 \]

### Step 3: Simplify the equation
First, we will simplify the numerator:
\[ x + (x + 1) + (x + 2) + (x + 5) + (x + 7) = 5x + (1 + 2 + 5 + 7) = 5x + 15 \]
Now, substituting this back into the mean equation gives us:
\[ \frac{5x + 15}{5} = 11 \]

### Step 4: Multiply both sides by 5
To eliminate the fraction, we multiply both sides by 5:
\[ 5x + 15 = 55 \]

### Step 5: Solve for \( x \)
Now, we isolate \( x \):
\[ 5x = 55 - 15 \]
\[ 5x = 40 \]
\[ x = \frac{40}{5} = 8 \]

### Step 6: Find the first three observations
Now that we have the value of \( x \), we can find the first three observations:
- \( x = 8 \)
- \( x + 1 = 9 \)
- \( x + 2 = 10 \)

### Step 7: Calculate the mean of the first three observations
Now, we calculate the mean of the first three observations:
\[ \text{Mean} = \frac{8 + 9 + 10}{3} = \frac{27}{3} = 9 \]

### Final Answer
The mean of the first three observations is \( 9 \).

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