Sum of the first 689 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 689 square numbers, you ask? Here we will give you the formula to calculate the first 689 square numbers and then we will show you how to calculate the first 689 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 689 square numbers, we enter n = 689 into our formula to get this:

		689(689 + 1) × (2(689) + 1)
		6

First, calculate each section of the numerator: 689(689 + 1) equals 475410 and (2(689) + 1) equals 1379. Therefore, the problem above becomes this:

		475410 × 1379
		6

Next, we calculate 475410 times 1379 which equals 655590390. Now our problem looks like this:

		655590390
		6

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

655590390 ÷ 6 = 109265065

There you go. The sum of the first 689 square numbers is 109265065.

You may also be interested to know that if you list the first 689 square numbers 1, 2, 9, etc., the 689th square number is 474721.

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