Arithmetic Progression first m + n terms

[sriram10]

If the sum of first m terms of an a.p is n and sum of first n terms is m,then show that sum of its first (m+n)terms is -(m+n).I have a doubt pls verify how to do this.

[Jonasmathwork]

Given S_m = n & S_n = m
=> S_m = m/2[2a + (m - 1)d] = n
Or
2ma + m(m - 1)d = 2n ---------- (1)
And
S_n = n/2 [2a + (n - 1)d] = m
Or
2na + n(n - 1)d = 2m ----------- (2)

Subtract (2) from (1) , we get

2ma - 2na + d[m (m - 1) - n (n - 1)] = 2 (n - m)
2a (m - n) + d [ m^2 - m - n^2 + n ] = 2 (n - m)
2a (m - n) + d [(m - n) (m + n) - (m -n)] = 2(n - m)
(m - n)[2a + d (m + n -1)] = 2(n - m)
2a + d (m + n - 1) = -2 (Cancel (m - n) from both sides) ------------------ (3)

Again S_m+n = (m +n) / 2 [2a + (m + n - 1)d]
= (m +n) / 2 * -2 (using equation (3)]

=> S_ m+n = -(m +n)

Previously i have also doubt on Arithmetic Progression , But thanks to all math web sites to help me a lot......