Sum off all numbers between x-y using algebra

Question

Easiest and most efficient way to calculate the total sum of all numbers between x and y?

for example, assume x is 5 and y is 30. what would all of the numbers between 5 and 30 be as a total sum?

Answer 1

Using Gauss' method directly, you would have

 

5   +  6 + ..... + 29 + 30 = SUM

30 + 29+.....    + 6 + 5   = SUM

_____________________________

35 + 35 + ..... + 35 + 35 = 2 * SUM

 

Note that there are (30-5)+1 = 26 numbers if you include the 5 and the 30.  We DO want both of them.

So, 26*35 = 2 * SUM.

 

Then Divide by 2 - because we want 1 sum and not 2 of them...

13+35 = SUM = 455.

 

Answer 2

First, you should know Gauss' very famous formula for the Sum of The Numbers from 1 to N:

 

    Sum of The Numbers from 1 to N = (N)*(N+1)/2

 

Consider the example of the Sum of 1+2+3+4+5:

  Let's add it twice:

         1   +   2    +   3    +   4   +   5   = Sum

         5   +   4    +   3    +   2   +   1   = Sum

       ---------------------------------------------

         6   +   6    +   6    +   6   +   6   = 2*Sum

 

There a five sixes in twice the Sum, so Sum = 5*6/2 = 15    [easily checked]

 

Now, the sum of the numbers from 5 to 30 (include both 5 and 30, that is, 5+6+7+8+,,,+29+30).  Well, that is the sum of the numbers from 1 to 30 (that is, 30*31/2) minus the sum of the numbers from 1 to 4 (that is, 4*5/2).

   For these numbers:  30*31/2 -  4*5/2

                                    15*31  -  2*5

                                  465 - 10

                                     455

 

p.s., I checked this using MS Excel.