((n+1)/2)n

Date: 01/31/2002 at 05:52:53
From: Alan Naude
Subject: Proof of an equation

If you want to figure out the total of a series of numbers in order, 
e.g.: 1+2+3+4+5+6+7+8+9 etc., you would use the formula below where 
n is the final number of your series.

((n+1)/2)n

Why, and what is the proof of the formula?

Date: 01/31/2002 at 08:28:51
From: Doctor Jerry
Subject: Re: Proof of an equation

Hi Alan,

Any finite sum of the form

S = a + (a+d) + (a+2d) + (a+3d) + ... + (a+(n-1)d)

can be summed. For the series you mentioned, take a = 1 and d = 1.

Write the sum as above and reverse it:

S = a + (a+d) + (a+2d) + (a+3d) + ... + (a+(n-1)d)

S = (a+(n-1)d) + (a+(n-2)d) + ...+(a+d) + a

Add together:

2S = n*[2a+(n-1)d]

So,

S = n[a + a + (n-1)d]/2.

So, to sum a finite arithmetic progression, average the first and last 
terms and multiply by the number n of terms.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/