Gauss's Invention of the Formula

Do U know how Gauss arrived at the formula n(n+1)/2 to get the sum of n
numbers??





Gauss was a very great mathematician who has contributed extensively to
the field of mathematic..



One day when he was doing 7th standard ...all his classmates were shouting in the class...



suddenly, the teacher came in and got so much irritated that he gave a
very tough punishment for them all....



The teacher wrote down a problem to find out the sum of "1+2+3+4+............10,0000".



When everybody started to work out this problem....



Mr. Gauss thought for a while and came up with the following Invention..





say



" 1 + 2 + 3 + 4+..............................................+N = S

(where S is the sum ) .............................Equation 1



N + (N-1) + (N-2) + (N-3 ) +............................................ + 1 = S (writing Equation 1in Reverse Order)...Equation 2



Adding the above two equations

(N+1)+ (N+1) + (N+1) + (N+1)+............................................ +(N+1) = 2S

equation 3



Now taking (N+1) outside as it is common to all the n terms, we get

(N+1)(1 + 1+ 1 +1................................................ .............................+1)= 2S




then


(N+1) * N = 2S ----------------------> S (Sum)= (N+1)*N/2

--------------------> Equation 4..





Now Gauss calculated S= (10000*10001/2)= 50005000 ......



within 5 minutes, Gauss calculated the sum and said to his teacher , "I
have found the answer and If u want U can cross check this".The teacher
got so much offended but could not disprove Gauss's solution