SOLUTION: Confused about problem. Need to solve this but I do not know how to use the formula. Need to know the sum of pairwise comparisons. 1+2+3+...+498+499+500

Question 5666: Confused about problem. Need to solve this but I do not know how to use the formula. Need to know the sum of pairwise comparisons.
1+2+3+...+498+499+500
Answer by prince_abubu(198) (Show Source): You can put this solution on YOUR website!
You might be looking for a formula that will add up the consecutive natural numbers from 1 to n. You know that f(1) = 1, f(2) = 1 + 2 = 3; f(3) = 1 + 2 + 3 = 6, etc... and ultimatley, f(n) = 1 + 2 + 3 + .... + n. But you don't want to keep on adding the consecutive terms because it'd take you forever, especially if n were big enough.

I've seen this one before and the derivation is quite clever. Look again at the expression that you wrote 1 + 2 + 3 + .... + 498 + 499 + 500. Imagine that each number were on a strip of long paper. Fold the paper exactly in half. This way, the 1 will be paired with the 500, the 2 will be paired with the 499, the 3 will be paired with the 498, and so on.

If you notice, each pair after the fold ALWAYS has a sum of 501. Aha! But how many pairs would there be? There would be 250 pairs because you folded the strip in half. So 1 + 2 + 3 + 4 + ... + 500 = 501 * 250 = 125250.

Working with our example here, N = 500, the last number in our summation. When we "paired the opposite ends", we found that the sums were 501, which really is N + 1. Each pair's sum was 501, but how many pairs were there? 250 pairs, which really is N/2.

Now, we can put together the formula written in nicer form. As mentioned, that formula actually equals 1 + 2 + 3 + ... + n.
RELATED QUESTIONS
X+9=-2 WHERE DO I START WITH THIS PROBLEM? I NEED TO KNOW ALL THE SSTEPS I USE TO SOLVE... (answered by longjonsilver)

I need to solve for x and y by using substitution. I am not sure of how to go about... (answered by ankor@dixie-net.com)

y = -x^2 + 7x - 2 I need to graph this quadratic function. I know I will need to... (answered by Fombitz)

I need to solve this linear equation, but i just do not know how. I need the proper steps (answered by tutorcecilia)

Factor and check by multiplying x^8 + 3x^7 just Factor: 81c^6 - 3d^6 Are... (answered by richwmiller)

I have this question I am extremely confused, I know I need to use differential equations (answered by Edwin McCravy)

Hi, could you please help me on how to solve this problem? Determine the area of the... (answered by adamchapman)

In a triangle three sides measure: 12, 18 and 24. What is the measure of the smallest... (answered by shree840,Alan3354)