Trignometry I think idk

Question

Can you show for every positive whole number n,

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

What about

1^2+2^2+...+n^2=n(n+1)(2n+1) all of it divided by 6

Answer 1

A) There was a great mathematician named Fredrick Gauss. One day in school, his teacher wanted to keep the students busy so she asked them to add all numbers from 1 to 100.

Gauss did it like this, he listed the numbers and then added the inverse grouping like this

1 2 3 4 5 ....................................99 100

100 99 98 97 96 .................................. 2 1 and then he added them together like so..

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

101 101 101 101 101 ................................ 101 101

Then he multiplied 101 x 100 to get 10100 but since he doubled the sum by adding the "backward" number, he must then divide by 2

So the answer is 10100/2 = 5050 If n = 100 and n+1 = 101 then n(n+1)/2 = 5050 !!

B) The second question is a little more involved but you can see the proof at http://mathforum.org/library/drmath/view/56920.html