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

Answer

To find the mean of the first three observations given that the mean of the observations \( x, x+2, x+4, x+6, \) and \( x+8 \) is 11, we can follow these steps:

### Step 1: Set up the equation for the mean
The mean of the observations is given by the formula:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Total number of observations}} \]

In our case, the observations are \( x, x+2, x+4, x+6, \) and \( x+8 \). Therefore, we can write:

\[ \frac{x + (x + 2) + (x + 4) + (x + 6) + (x + 8)}{5} = 11 \]

### Step 2: Simplify the equation
Now, let's simplify the sum in the numerator:

\[ x + (x + 2) + (x + 4) + (x + 6) + (x + 8) = 5x + (2 + 4 + 6 + 8) = 5x + 20 \]

So the equation becomes:

\[ \frac{5x + 20}{5} = 11 \]

### Step 3: Eliminate the denominator
To eliminate the denominator, multiply both sides by 5:

\[ 5x + 20 = 55 \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \) by isolating it:

\[ 5x = 55 - 20 \]
\[ 5x = 35 \]
\[ x = 7 \]

### Step 5: Find the first three observations
Now that we have \( x \), we can find the first three observations:

1. The first observation: \( x = 7 \)
2. The second observation: \( x + 2 = 7 + 2 = 9 \)
3. The third observation: \( x + 4 = 7 + 4 = 11 \)

So the first three observations are \( 7, 9, \) and \( 11 \).

### Step 6: Calculate the mean of the first three observations
Now we can calculate the mean of these three observations:

\[ \text{Mean} = \frac{7 + 9 + 11}{3} \]
\[ \text{Mean} = \frac{27}{3} = 9 \]

### Final Answer
The mean of the first three observations is **9**.

---