The mean of `x,x+2,x+4,x+6` and `x+8` is 11, find the mean of the first three observations.

Answer

To solve the problem step by step, we will follow the process of finding the mean of the observations and then calculating the mean of the first three observations.

### Step 1: Write down the observations
The observations given are:
- \( x \)
- \( x + 2 \)
- \( x + 4 \)
- \( x + 6 \)
- \( x + 8 \)

### Step 2: Calculate the mean of the observations
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Total number of observations}} \]
Here, the total number of observations is 5.

### Step 3: Find the sum of the observations
Now, we will find the sum of the observations:
\[ \text{Sum} = x + (x + 2) + (x + 4) + (x + 6) + (x + 8) \]
Combining like terms, we get:
\[ \text{Sum} = x + x + 2 + x + 4 + x + 6 + x + 8 = 5x + 20 \]

### Step 4: Set up the equation for the mean
According to the problem, the mean is given as 11:
\[ \frac{5x + 20}{5} = 11 \]

### Step 5: Solve for \( x \)
To eliminate the fraction, multiply both sides by 5:
\[ 5x + 20 = 55 \]
Now, subtract 20 from both sides:
\[ 5x = 55 - 20 \]
\[ 5x = 35 \]
Now, divide by 5:
\[ x = \frac{35}{5} = 7 \]

### Step 6: Find the first three observations
Now that we have \( x = 7 \), we can find the first three observations:
- First observation: \( x = 7 \)
- Second observation: \( x + 2 = 7 + 2 = 9 \)
- Third observation: \( x + 4 = 7 + 4 = 11 \)

### Step 7: Calculate the mean of the first three observations
Now, we calculate the mean of the first three observations:
\[ \text{Mean of first three observations} = \frac{7 + 9 + 11}{3} \]
Calculating the sum:
\[ 7 + 9 + 11 = 27 \]
Now, divide by 3:
\[ \text{Mean} = \frac{27}{3} = 9 \]

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

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