Let the sequence a1,a2,a3.........  form an A.P. Then a12−a22+a32−a42+........+a2n−12−a2n2=

a12−a22+a32−a42+........+a2n−12−a2n2 =(a1−a2)(a1+a2)+(a3−a4)(a3+a4)+.............+(a2n−1−a2n)(a2n−1+a2n) =−d[a1+a2+...+a2n] =−d2n2[a1+a2n] =−dn[a12−a2n2]a1−a2n=−nd(a12−a2n2)−(2n−1)d =n2n−1(a12−a2n2)