The average of `1^2 , 2^2 , 3^2 , 4^2 , 5^2 , 6 ^2 , 7^2 , 8^2 , 9^2 , 10^2` is :

Options

A. 42.5
B. 46.5
C. 50.5
D. 38.5

Answer

To find the average of the squares of the first ten natural numbers, we can follow these steps:

### Step 1: Identify the numbers
We need to calculate the squares of the numbers from 1 to 10. The numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

### Step 2: Calculate the squares
Now, we calculate the squares of each of these numbers:
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)
- \(7^2 = 49\)
- \(8^2 = 64\)
- \(9^2 = 81\)
- \(10^2 = 100\)

### Step 3: Sum the squares
Next, we sum all these squared values:
\[ 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 \]

Calculating this step-by-step:
- \(1 + 4 = 5\)
- \(5 + 9 = 14\)
- \(14 + 16 = 30\)
- \(30 + 25 = 55\)
- \(55 + 36 = 91\)
- \(91 + 49 = 140\)
- \(140 + 64 = 204\)
- \(204 + 81 = 285\)
- \(285 + 100 = 385\)

So, the total sum of the squares is \(385\).

### Step 4: Count the number of observations
The total number of observations (the numbers from 1 to 10) is \(10\).

### Step 5: Calculate the average
Now, we can calculate the average using the formula:
\[ \text{Average} = \frac{\text{Total Sum}}{\text{Number of Observations}} = \frac{385}{10} \]
Calculating this gives:
\[ \text{Average} = 38.5 \]

### Final Answer
Thus, the average of \(1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2\) is \(38.5\).

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