Previous Year Questions

To find the average of the squares of the first 45 natural numbers, we use the formula for the sum of squares of the first nnn natural numbers:

Sn=1²+2²+3²+⋯+n²=n(n+1)(2n+1)/6

For n=45, the sum of the squares is:

S45=45(45+1)(2 X 45+1)6=45 X 46 X 91/6=31395

Thus, the sum of the squares of the first 45 natural numbers is 31395.

Step 2: Find the average

The average of the squares is the sum of the squares divided by the number of terms (which is 45):

Average=31395/45=697.67

Final Answer: The average of the squares of the first 45 natural numbers is 697.67