What is the average of the squares of the first 19 natural numbers?

Options

A. 124
B. 127.5
C. 130
D. 133.5

Answer

To find the average of the squares of the first 19 natural numbers, we can follow these steps:

### Step 1: Identify the first 19 natural numbers
The first 19 natural numbers are:
1, 2, 3, ..., 19.

### Step 2: Use the formula for the sum of squares
The formula for the sum of the squares of the first n natural numbers is given by:
\[ \text{Sum of squares} = \frac{n(n + 1)(2n + 1)}{6} \]
For our case, \( n = 19 \).

### Step 3: Substitute n into the formula
Now we substitute \( n = 19 \) into the formula:
\[ \text{Sum of squares} = \frac{19(19 + 1)(2 \cdot 19 + 1)}{6} \]
This simplifies to:
\[ \text{Sum of squares} = \frac{19 \cdot 20 \cdot 39}{6} \]

### Step 4: Calculate the sum of squares
Now we calculate the product:
\[ 19 \cdot 20 = 380 \]
\[ 2 \cdot 19 + 1 = 39 \]
Now, multiply:
\[ 380 \cdot 39 = 14820 \]
Now divide by 6:
\[ \text{Sum of squares} = \frac{14820}{6} = 2470 \]

### Step 5: Calculate the average
The average is calculated by dividing the sum of squares by the number of terms (which is 19):
\[ \text{Average} = \frac{\text{Sum of squares}}{n} = \frac{2470}{19} \]

### Step 6: Perform the division
Now we perform the division:
\[ \frac{2470}{19} = 130 \]

### Final Answer
Thus, the average of the squares of the first 19 natural numbers is **130**.

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