The average of square of natural numbers from 1 to 71 is ?

Options

A. 1616
B. 1716
C. 1728
D. 1692

Answer

To find the average of the squares of natural numbers from 1 to 71, we can follow these steps:

### Step 1: Understand the Formula for the Average of Squares
The average of the squares of the first \( n \) natural numbers can be calculated using the formula:
\[ \text{Average} = \frac{1^2 + 2^2 + 3^2 + \ldots + n^2}{n} \]
Where \( n \) is the last natural number in the range.

### Step 2: Identify \( n \)
In this case, \( n = 71 \).

### Step 3: Use the Formula for the Sum of Squares
The sum of the squares of the first \( n \) natural numbers is given by the formula:
\[ \sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6} \]
Substituting \( n = 71 \):
\[ \sum_{k=1}^{71} k^2 = \frac{71(71 + 1)(2 \times 71 + 1)}{6} \]

### Step 4: Calculate Each Component
1. Calculate \( 71 + 1 = 72 \)
2. Calculate \( 2 \times 71 + 1 = 142 + 1 = 143 \)

### Step 5: Substitute and Calculate the Sum
Now substitute these values into the sum of squares formula:
\[ \sum_{k=1}^{71} k^2 = \frac{71 \times 72 \times 143}{6} \]

### Step 6: Perform the Multiplication
1. Calculate \( 71 \times 72 = 5112 \)
2. Now multiply \( 5112 \times 143 \):
\[ 5112 \times 143 = 731856 \]

### Step 7: Divide by 6 to Get the Sum of Squares
Now divide by 6:
\[ \frac{731856}{6} = 121976 \]

### Step 8: Calculate the Average
Finally, to find the average of the squares, divide the sum of squares by \( n \):
\[ \text{Average} = \frac{121976}{71} = 1716 \]

### Conclusion
Thus, the average of the squares of natural numbers from 1 to 71 is **1716**.

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