How much will be the average of the squares of natural numbers from 1 to 35?

Options

A. a. 612.5
B. b. 1225
C. c. 426
D. d. 324

Answer

To find the average of the squares of natural numbers from 1 to 35, we can follow these steps:

### Step 1: Identify the formula for the sum of squares
The formula for the sum of the squares of the first \( n \) natural numbers is given by:
\[ S = \frac{n(n + 1)(2n + 1)}{6} \]
where \( n \) is the largest natural number in the range, which in this case is 35.

### Step 2: Substitute \( n \) into the formula
Substituting \( n = 35 \) into the formula:
\[ S = \frac{35(35 + 1)(2 \cdot 35 + 1)}{6} \]
Calculating \( 35 + 1 \) and \( 2 \cdot 35 + 1 \):
\[ S = \frac{35 \cdot 36 \cdot 71}{6} \]

### Step 3: Calculate the product
Now, calculate the product:
\[ 35 \cdot 36 = 1260 \]
Then, multiply by 71:
\[ 1260 \cdot 71 = 89460 \]

### Step 4: Divide by 6 to find the sum of squares
Now, divide by 6 to find the sum of squares:
\[ S = \frac{89460}{6} = 14910 \]

### Step 5: Calculate the average
To find the average of the squares, we divide the sum of squares by the number of natural numbers (which is 35):
\[ \text{Average} = \frac{S}{n} = \frac{14910}{35} \]

### Step 6: Perform the division
Now, perform the division:
\[ \text{Average} = 426 \]

### Final Answer
The average of the squares of natural numbers from 1 to 35 is \( 426 \).

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