Quadratic Gauss sum
Okay kiddo, have you ever counted how many numbers there are from 1 to 10? That's 10 numbers, right? What about if I asked you to add up all those numbers, from 1 to 10? Do you know how to do that? Well, one way is to start with 1, then add 2 to it to get 3, then add 3 to get 6, and keep going until you finally add 10 to get 55.
Now imagine I asked you to add up all the squares of those numbers, from 1 to 10. So instead of adding 1+2+3+...+10, you're adding 1^2+2^2+3^2+...+10^2. That's a bit trickier, right? But don't worry, there's a special trick we can use called the quadratic Gauss sum.
Basically, the quadratic Gauss sum is a formula that tells us how to add up all the squares of numbers in a certain pattern. It looks like this:
S = (1^2 + 2^2 + ... + (n-1)^2 + n^2)
S = (n/3) x [(2n+1)(n+1)]
Let me break it down for you. First, we find the sum of all the squares of numbers from 1 to n, just like we did before. Then we plug n into the formula:
S = (n/3) x [(2n+1)(n+1)]
The n/3 part is just a coefficient that helps us simplify things. The (2n+1) and (n+1) parts are just expressions that give us the answer.
So if we wanted to find the sum of the squares of numbers from 1 to 10, we would do this:
S = (10/3) x [(2x10+1)(10+1)]
S = (10/3) x [21 x 11]
S = (10/3) x 231
S = 770
And there you have it, kiddo! The sum of the squares of numbers from 1 to 10 is 770. Pretty cool, huh?
Now imagine I asked you to add up all the squares of those numbers, from 1 to 10. So instead of adding 1+2+3+...+10, you're adding 1^2+2^2+3^2+...+10^2. That's a bit trickier, right? But don't worry, there's a special trick we can use called the quadratic Gauss sum.
Basically, the quadratic Gauss sum is a formula that tells us how to add up all the squares of numbers in a certain pattern. It looks like this:
S = (1^2 + 2^2 + ... + (n-1)^2 + n^2)
S = (n/3) x [(2n+1)(n+1)]
Let me break it down for you. First, we find the sum of all the squares of numbers from 1 to n, just like we did before. Then we plug n into the formula:
S = (n/3) x [(2n+1)(n+1)]
The n/3 part is just a coefficient that helps us simplify things. The (2n+1) and (n+1) parts are just expressions that give us the answer.
So if we wanted to find the sum of the squares of numbers from 1 to 10, we would do this:
S = (10/3) x [(2x10+1)(10+1)]
S = (10/3) x [21 x 11]
S = (10/3) x 231
S = 770
And there you have it, kiddo! The sum of the squares of numbers from 1 to 10 is 770. Pretty cool, huh?