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General Approach for Sum of Arithmetic Series

Date: 08/24/2006 at 17:46:55
From: robert 
Subject: algebra/geometry

What is the formula for figuring out the sum of adding consecutive 
even numbers?  2 + 4 = 6, 2 + 4 + 6 = 12, 2 + 4 + 6 + 8 = 20, etc.

Date: 08/25/2006 at 14:54:39
From: Doctor Greenie
Subject: Re: algebra/geometry

Hi, Robert -

You can memorize the formula for this particular type of problem; or
you can learn a (fairly easy) general method for finding the sum of
ANY sequence of numbers in which the difference between successive
numbers is a constant amount.

The quick formulas for special cases, including yours, are the 

(1) Sum of first n odd integers:  n^2

  1 = 1 = 1^2
  1+3 = 4 = 2^2
  1+3+5 = 9 = 3^2
  1+3+5+7 = 16 = 4^2

(2) Sum of first n even integers:  n^2 + n;  or  n(n + 1)

  2 = 1(2)
  2+4 = 6 = 2(3)
  2+4+6 = 12 = 3(4)
  2+4+6+8 = 20 = 4(5)

(3) Sum of first n integers:  (n(n + 1))/2

  1 = 1(2)/2
  1+2 = 3 = 2(3)/2
  1+2+3 = 6 = 3(4)/2
  1+2+3+4 = 10 = 4(5)/2

All of these are special cases of a general formula which I think is 
easy to remember:

  Sum = (number of numbers) * (first + last)/2

In the second part of this formula, we are adding two numbers and 
dividing by 2; that means we are taking the average of the two 
numbers.  So I like to think of this formula as

  Sum = (number of numbers) * (average of first and last)

In any sequence of numbers in which the difference between successive 
terms is the same, the average of all the numbers is the same as the 
average of the first and last numbers.  So our formula really just 
says that the sum of a bunch of numbers is equal to the average, 
multiplied by how many there are.  This makes perfect sense, since an 
average is found by dividing the total by how many there are.

Let's look at a couple of the special case formulas and see that they 
are indeed particular examples of this general formula.

(2 again) Sum of first n even integers


There are n numbers; the average of the first and last is

  (2n+2)/2 = n+1

And so our general formula gives us the special case formula:

  S = n(n+1)

(3 again) Sum of first n integers


There are again n numbers; the average of the first and last is 
(n+1)/2.  The general formula again gives us the special case formula:

  S = n(n+1)/2

It certainly doesn't hurt to memorize the special case formulas.  But
you can use the general formula to find sums of vastly different 
sequences.  For example...


The difference between the first and last terms is 240 - 90 = 150; the 
common difference between terms is 5; so there are 150/5 = 30 terms
after the first one; so the number of terms in the sequence is 31.

The average of the first and last terms is (90+240)/2 = 330/2 = 165.

So the sum of this sequence of numbers is

  31(165) = 5115

I hope this helps.  Please write back if you have any further 
questions about any of this.

- Doctor Greenie, The Math Forum
Associated Topics:
High School Sequences, Series

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