how to find the summ

Question

Find the sum: 1+2+3=...+94 using Gauss's approach Please show how you arrive at this answer

Answer 1

Hi Henriettas,

To use Gauss' approach split the numbers you want to add into two even groups.  In this case you have 94 sequential numbers to add, so you will have two groups of 47.  To make the method easier to visualize, write the first group with the numbers going up (increasing) and below that write the second group with the numbers coming down (decreasing):

 1 +   2 +   3 + ... + 45 + 46 + 47

94 + 93 + 92 + ... + 50 + 49 + 48

Notice that if you add each column of numbers, you get 95.  1 + 94 = 95, 2 + 93 = 95, ... 47 + 48 = 95.  This means you have 47 sets of numbers that add to 95, so the original sum is 47*95 = 4465.

If you end up working on a problem with an odd number of terms to be added, just leave off the 1 at the beginning -- then use Gauss' method on the remaining numbers.  Just don't forget to add that 1 back in at the end to get your final answer!

Answer 2

       1
    + 2
    + 3
    + 4
    + 5
94 + 6  = 100
93 + 7  = 100
92 + 8  = 100
91 + 9  = 100
90 + 10 = 100
89 + 11 = 100
88 + 12 = 100
..................
51 + 49 = 100
     + 50
___________________
4 * 1000 + 450  + 15 = 4465