DISCRETE MATHEMATICS QUESTION Prove that 12 22 32 42 n n n1

DISCRETE MATHEMATICS QUESTION

Prove that 12 22+ 32- 42+ n? n (n+1)

Case 1. n is even,n=2m

1^2-2^2+3^2-4^2+...(2m-1)^2.-(2m)^2

(1-2)(1+2)+(3-4)(3+4)...+(2m-1-2m)(2m-1+2m)

=-(1+2)-(3+4)-......-(2m-1+2m)

=-1-2-3-4....-(2m-1+2m)

This is sum of natural numbers from 1 to n

Hence, -n(n+1)/2

Case 2. n is odd,n=2m+1

1^2-2^2+3^2-4^2+...(2m-1)^2.-(2m)^2+(2m+1)^2

(1-2)(1+2)+(3-4)(3+4)...+(2m-1-2m)(2m-1+2m)

=-(1+2)-(3+4)-......-(2m-1+2m)

=-1-2-3-4....-(2m-1+2m)+(2m+1)^2

=-(n-1)n/2+n^2

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

=n(n+1)/2