How does this summation equal \frac{n(n+1)(2n+1)}{6}?

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In summary, the conversation was about a summation involving the square of a variable "i" starting from 1 to "n". The equation for this summation was given as \frac{n(n+1)(2n+1)}{6}. The person had attempted to find a solution by writing out the summation in reverse and adding it to the original summation, but was unable to fully explain the pattern. They were looking for a link or explanation as to why this equation is the solution.
  • #1
vanmaiden
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Homework Statement


I'm not very proficient with LAtex, so I'll try to translate this mess the best I can. It's a summation
[itex]\sum[/itex] i2 (on the bottom, there would be an "i = 1") (on the top, there would be an "n") this summation equals [itex]\frac{n(n+1)(2n+1)}{6}[/itex]

In the summation, basically, "i" starts off at one and goes to "n"

Why does this summation equal [itex]\frac{n(n+1)(2n+1)}{6}[/itex]?

Homework Equations


The definition of a definite integral? I found this summation through doing definite integrals using Riemann sums. I found the answer to the summation online, but I wanted to know how one arrives at the answer.

The Attempt at a Solution


So far, I have done this:

S = 12 + 22 + 32 + ... + (n - 1)2 + n2
S = n2 + (n - 1)2 + (n - 2)2 + ... + 22 + 12

I just wrote out a part of the summation, and under it, I did the same summation part but in reverse.

Then...I added the two summations to get

2S = n2 + 1 + (n - 1)2 + 4 + (n - 2)2 + 9 + ... + (n - 1)2 + 4 + n2 + 1

I saw somewhat of a pattern, but it was hard to explain and I had to stop here. If someone could give me a link to an explanation, or if someone could fit an explanation in a response or two as to why this summation equals [itex]\frac{n(n+1)(2n+1)}{6}[/itex], that would be fantastic. Thank you in advance!
 
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  • #2
Try this: http://www.trans4mind.com/personal_development/mathematics/series/sumNaturalSquares.htm"
 
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What is "Summation of i Squared"?

"Summation of i Squared" is a mathematical concept that involves adding up the squares of a series of numbers, starting from 1 up to a given number. This is represented by the symbol Σi^2, where i is the variable representing the numbers being squared and summed.

What is the formula for calculating "Summation of i Squared"?

The formula for "Summation of i Squared" is Σi^2 = (n*(n+1)*(2n+1))/6, where n is the given number up to which the squares are being summed.

Why is "Summation of i Squared" important?

"Summation of i Squared" has many applications in mathematics and science, such as in calculating the area under a curve, finding the sum of squares of deviations in statistics, and in solving certain differential equations. It also helps in understanding the concept of series and sequences.

What is the difference between "Summation of i" and "Summation of i Squared"?

"Summation of i" involves adding up a series of numbers from 1 to a given number, while "Summation of i Squared" involves adding up the squares of the same series of numbers. In other words, "Summation of i Squared" is the sum of the first n square numbers, while "Summation of i" is the sum of the first n natural numbers.

What is the significance of the value of "Summation of i Squared"?

The value of "Summation of i Squared" can provide information about the behavior of a series of numbers, such as whether it is converging or diverging. It can also be used to calculate other important values, such as the mean and variance in statistics, and can help in solving certain mathematical problems and equations.

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