What is the mean of the squares of the first 20 natural numbers ?

Options

A. 151.5
B. 143.5`
C. 65
D. 72

Explanation

To find the mean of the squares of the first 20 natural numbers, we can follow these steps:

### Step 1: Identify the first 20 natural numbers
The first 20 natural numbers are:
1, 2, 3, ..., 20.

### Step 2: Calculate the squares of these numbers
We need to find the squares of each of these numbers:
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)
- \(7^2 = 49\)
- \(8^2 = 64\)
- \(9^2 = 81\)
- \(10^2 = 100\)
- \(11^2 = 121\)
- \(12^2 = 144\)
- \(13^2 = 169\)
- \(14^2 = 196\)
- \(15^2 = 225\)
- \(16^2 = 256\)
- \(17^2 = 289\)
- \(18^2 = 324\)
- \(19^2 = 361\)
- \(20^2 = 400\)

### Step 3: Use the formula for the sum of squares
The sum of the squares of the first \(n\) natural numbers can be calculated using the formula:
\[ \text{Sum} = \frac{n(n + 1)(2n + 1)}{6} \]
For \(n = 20\):
\[ \text{Sum} = \frac{20(20 + 1)(2 \cdot 20 + 1)}{6} = \frac{20 \cdot 21 \cdot 41}{6} \]

### Step 4: Calculate the sum
Calculating the sum step by step:
1. Calculate \(20 \cdot 21 = 420\)
2. Calculate \(2 \cdot 20 + 1 = 41\)
3. Now, calculate \(420 \cdot 41 = 17220\)
4. Finally, divide by 6:
\[ \text{Sum} = \frac{17220}{6} = 2870 \]

### Step 5: Calculate the mean
The mean is calculated by dividing the sum of squares by the number of observations (which is 20):
\[ \text{Mean} = \frac{\text{Sum}}{n} = \frac{2870}{20} = 143.5 \]

### Final Answer
The mean of the squares of the first 20 natural numbers is **143.5**.

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