What is the mean of the squares of the first 20 natural numbers?
(a)151.5
(b)143.5
(c)65
(d)72

Options

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

Answer

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

### Step 1: Understand the formula for the sum of squares
The formula for the sum of the squares of the first \( n \) natural numbers is given by:
\[ \text{Sum of squares} = \frac{n(n + 1)(2n + 1)}{6} \]
where \( n \) is the number of natural numbers.

### Step 2: Substitute \( n = 20 \)
For our case, since we are looking for the first 20 natural numbers, we will substitute \( n = 20 \) into the formula:
\[ \text{Sum of squares} = \frac{20(20 + 1)(2 \times 20 + 1)}{6} \]

### Step 3: Calculate the components
Now, calculate each part:
- \( n + 1 = 20 + 1 = 21 \)
- \( 2n + 1 = 2 \times 20 + 1 = 40 + 1 = 41 \)

### Step 4: Substitute back into the formula
Now substitute these values back into the sum of squares formula:
\[ \text{Sum of squares} = \frac{20 \times 21 \times 41}{6} \]

### Step 5: Calculate the product
Now calculate the product:
\[ 20 \times 21 = 420 \]
\[ 420 \times 41 = 17220 \]

### Step 6: Divide by 6
Now, divide this product by 6 to get the sum of squares:
\[ \text{Sum of squares} = \frac{17220}{6} = 2870 \]

### Step 7: Calculate the mean
To find the mean of the squares, we divide the sum of squares by the number of observations (which is 20):
\[ \text{Mean} = \frac{2870}{20} = 143.5 \]

### Final Result
Thus, the mean of the squares of the first 20 natural numbers is:
\[ \boxed{143.5} \]