Find the average of the squares of the first 45 natural numbers.

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Let the first 45 natural numbers be 1, 2, 3, ..., 45. The sum of the squares of the first n natural numbers is given by the formula: S = n(n+1)(2n+1)/6. Here, n = 45. So, S = 45 × 46 × 91 / 6. Let's calculate step by step:

First, 45 × 46 = 2070.
Then, 2070 × 91 = 188370.
Now, divide by 6: 188370 / 6 = 31395.

The average = Total sum / Number of terms = 31395 / 45 = 697.666... ≈ 697.67.

However, let's check the options. The closest option to 697.67 is option 3, but option 2 is 699.67. Let's double-check the calculation:

Sum = n(n+1)(2n+1)/6 = 45 × 46 × 91 / 6 = 2070 × 91 / 6 = 188370 / 6 = 31395.
Average = 31395 / 45 = 697.666... ≈ 697.67.

But the options do not match this value exactly. However, based on calculation, the correct answer is 697.67, which is option 3. But as per your answer key, option 2 is marked as correct. There may be a typographical error in the options. Based on the calculation, the correct answer is 697.67 (option 3).

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