Question

The arithmetic mean of \(4^2,\,8^2,\,12^2,\ldots,32^2\) is

• A. \(816\)
• B. \(612\)
• C. \(408\)
• D. \(204\)

Hint:

Arithmetic mean is calculated as \[ \boxed{\text{Mean}=\frac{\text{Sum of all observations}}{\text{Number of observations}}.} \] Always compute the total first and then divide by the number of terms.

Answer 1

The Correct Option is C

Step 1: Write the sequence. \[ 4^2,\;8^2,\;12^2,\;16^2,\;20^2,\;24^2,\;28^2,\;32^2 \] There are \[ n=8 \] terms.

Step 2: Evaluate the squares. \[ 16,\;64,\;144,\;256,\;400,\;576,\;784,\;1024 \] Their sum is \[ 16+64+144+256+400+576+784+1024=3264. \]

Step 3: Find the arithmetic mean. \[ \text{Mean} = \frac{3264}{8} = 408. \]

Step 4: Final conclusion. \[ \boxed{408} \] Hence, the correct option is \(\boxed{(C)}\).