Assertion : If the mean of five observations x, x + 2, x + 4, x + 6, x + 8 is 11, then mean of last three observations is 8.
Reason : Mean of n observations is equal to Sum of observations upon Number of observations

Options

A. If both assertion and reason are true and reason is the correct explanation of assertion.
B. If both assertion and reason are true but reason is not the correct explanation of assertion.
C. If assertion is true but reason is false.
D. If assertion is false but reason is true.

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question step by step.

### Step 1: Calculate the Mean of the Five Observations
The five observations are:
- \( x \)
- \( x + 2 \)
- \( x + 4 \)
- \( x + 6 \)
- \( x + 8 \)

The mean (average) of these observations is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Substituting the values:
\[ \text{Mean} = \frac{x + (x + 2) + (x + 4) + (x + 6) + (x + 8)}{5} \]

### Step 2: Simplify the Sum
Now, let's simplify the sum in the numerator:
\[ \text{Sum} = x + x + 2 + x + 4 + x + 6 + x + 8 = 5x + (2 + 4 + 6 + 8) = 5x + 20 \]

### Step 3: Set the Mean Equal to 11
According to the assertion, the mean is 11:
\[ \frac{5x + 20}{5} = 11 \]

### Step 4: Solve for \( x \)
Multiply both sides by 5:
\[ 5x + 20 = 55 \]
Now, subtract 20 from both sides:
\[ 5x = 35 \]
Finally, divide by 5:
\[ x = 7 \]

### Step 5: Find the Last Three Observations
The last three observations are:
- \( x + 4 = 7 + 4 = 11 \)
- \( x + 6 = 7 + 6 = 13 \)
- \( x + 8 = 7 + 8 = 15 \)

Thus, the last three observations are 11, 13, and 15.

### Step 6: Calculate the Mean of the Last Three Observations
Now, we calculate the mean of these last three observations:
\[ \text{Mean} = \frac{11 + 13 + 15}{3} \]

### Step 7: Simplify the Mean Calculation
Calculating the sum:
\[ 11 + 13 + 15 = 39 \]
Now, divide by 3:
\[ \text{Mean} = \frac{39}{3} = 13 \]

### Conclusion
The mean of the last three observations is 13, not 8. Therefore, the assertion is false, and the reason is true.

### Final Answer
- Assertion: False
- Reason: True

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