If a1,a2,a3,...........  are in A.P and a12−a22+a32-a42+............+a2k-12−a2k2=M(a12-a2k2) .then M=

We havea2−a1=a3−a2=...............a2k−a2k−1=dHence,a12−a22=(a1−a2)(a1+a2)=−d(a1+a2)   a32−a42=(a3−a4)(a3+a4)=−d(a3+a4)………………………..…………………………….a2k-12−a2k2=(a2k−1−a2k)(a2k−1+a2k)=-da2k−1+a2k∴a12−a22+a32-a42+............+a2k-12−a2k2=-da1+a2+a3+.......+a2k                                                                        =−d.2k2(a1+a2k)=−dk(a1+a2k)but  a2k=a1+(2k−1)d⇒−d=a1−a2k2k−1∴  the required sum =k2k−1(a1 2−a2k 2)⇒M=k2k−1