prove sum(x)=n(n+1)/2 from x=1 to x=n

Answer 1

what you have is
1+2+3+...+n

let S(n) = 1+2+3+...+ (n-2) + (n-1) + n
writing this in reverse order
S(n) = n + (n-1) + (n-2)+ ... + 3+2+1
add them
2S(n) = (1+n) + (2+n-1) + (3+n-2) +...+(3+n-2) + (2+n-1) + (n+1)
= (n/2)(n+1)
or
= n(n+1)/2 , as required

Answer 2

thank you Reiny