Question

Sum to (2n+1)terms the series 1-2+3-4+.......

Answer 1

1-2+3-4+5-6  +7..upto (2n+1) terms
=(1+3+5+7....n+1 terms)  - (2+4+6+....n terms)
=[(1+2+3+4+5+....2n+1 terms)  - (2+4+6+....n terms)] - (2+4+6+....n terms) 
=  (1+2+3+4+5+....2n+1 terms) - 2x(2+4+6+....n terms)
= (1+2+3+4+5+....2n+1 terms) - 2x2x(1+2+3+4+5+....n terms)
= [(2n+1)(2n+2) / 2 ]   - [4 x n(n+1) / 2]
 = (2n+1)(n+1)  - 2n(n+1)  = (n+1)[(2n+1) - 2n] = (n+1)
                               in terms of (2n+1), (n+1) = [(2n+1)/2] - 1/2  

Answer 2

Given series is
1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10 ..............

Consider the terms in pairs

(1 - 2) + (3 - 4) + (5 - 6) + (7 - 8) + (9 - 10) ..............

-1 + -1 + -1 + -1 + -1 .........

Sum of these terms for (2n+1) terms is –(2n+1) = –2n – 1