Sum of Numbers from 1 to n
Date: 01/09/2003 at 03:43:03 From: Kobina Subject: Summation of numbers Hi Dr. Math, Do you know if there is a formula for calculating the summation of numbers from 1 through n? For example, let's say I want to add the numbers from 1 through 290. Regards, Kobina
Date: 01/09/2003 at 06:32:21 From: Doctor Jeremiah Subject: Re: Summation of numbers Hi Kobina, Here is an easy way to remember how to do this: Say S is the sum of the numbers from 1 to n. That means: S = 1 + 2 + 3 + ... + n-2 + n-1 + n (n items) But S could alos be the sum of the numbers in reverse order: S = n + n-1 + n-2 + ... + 3 + 2 + 1 (n items) But what would happen if we added these two things together: S = 1 + 2 + 3 + ... + n-2 + n-1 + n S = n + n-1 + n-2 + ... + 3 + 2 + 1 =============================================== 2S = n+1 + n+1 + n+1 + ... + n+1 + n+1 + n+1 (n items) Now think about how multiplication is defined. 2 + 2 + 2 is the same as 3 x 2 4 + 4 + 4 + 4 + 4 + 4 is the same as 6 x 4 n+1 + n+1 + n+1 is the same as 3(n+1) And n+1 added to itself n times is n(n+1) So that means: 2S = n+1 + n+1 + n+1 + ... + n+1 + n+1 + n+1 (n items) is the same as: 2S = n(n+1) Now we solve for S (which is the sum we want) and we get: S = n(n+1)/2 Let me know if you have more questions about this. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/
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