The average of the squares of the first ten natural numbers is

Options

A. 40
B. 50
C. 47.5
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 first ten natural numbers
The first ten natural numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

### Step 2: Calculate the squares of these numbers
Now, we will 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 will sum these squares:
\[ 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 \]
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 sum of the squares is \(385\).

### Step 4: Calculate the average
The average is calculated by dividing the sum of the squares by the number of terms. Here, we have 10 terms:
\[ \text{Average} = \frac{\text{Sum of squares}}{\text{Number of terms}} = \frac{385}{10} = 38.5 \]

### Final Answer
The average of the squares of the first ten natural numbers is **38.5**.

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