Sum of the first 684 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 684 square numbers, you ask? Here we will give you the formula to calculate the first 684 square numbers and then we will show you how to calculate the first 684 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 684 square numbers, we enter n = 684 into our formula to get this:

		684(684 + 1) × (2(684) + 1)
		6

First, calculate each section of the numerator: 684(684 + 1) equals 468540 and (2(684) + 1) equals 1369. Therefore, the problem above becomes this:

		468540 × 1369
		6

Next, we calculate 468540 times 1369 which equals 641431260. Now our problem looks like this:

		641431260
		6

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

641431260 ÷ 6 = 106905210

There you go. The sum of the first 684 square numbers is 106905210.

You may also be interested to know that if you list the first 684 square numbers 1, 2, 9, etc., the 684th square number is 467856.

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