Sum of the first 690 square numbers

We define square numbers as numbers that when squared will equal a whole number. Thus, the list of the first square numbers starts with 1, 4, 9, 16, and so on.

What is the sum of the first 690 square numbers, you ask? Here we will give you the formula to calculate the first 690 square numbers and then we will show you how to calculate the first 690 square numbers using the formula.

The formula to calculate the first n square numbers is displayed below:

		n(n + 1) × (2(n) + 1)
		6

To calculate the sum of the first 690 square numbers, we enter n = 690 into our formula to get this:

		690(690 + 1) × (2(690) + 1)
		6

First, calculate each section of the numerator: 690(690 + 1) equals 476790 and (2(690) + 1) equals 1381. Therefore, the problem above becomes this:

		476790 × 1381
		6

Next, we calculate 476790 times 1381 which equals 658446990. Now our problem looks like this:

		658446990
		6

Finally, divide the numerator by the denominator to get our answer:

658446990 ÷ 6 = 109741165

There you go. The sum of the first 690 square numbers is 109741165.

You may also be interested to know that if you list the first 690 square numbers 1, 2, 9, etc., the 690th square number is 476100.

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