If the sequence `a_1, a_2, a_3,....... a_n ,dot` forms an A.P., then prove that `a_1^2-a_2^2+a_3^2-a_4^2+.......+ a_(2n-1)^2 - a_(2n)^2=n/(2n-1)(a_1^2-a_(2n)^2)`

Explanation

Let d be the common difference of AP.
`LHS=a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+"......."+a_(2n-1)^(2)-a_(2n)^(2)`
`=(a_(1)-a_(2))(a_(1)+a_(2))+(a_(3)-a_(4))(a_(3)+a_(4))+"......."+(a_(2n-1)-a_(2n))(a_(2n-1)+a_(2n))`
`=-d(a_(1)+a_(2)+"......"+a_(2n-1)+a_(2n))`
`=-d[(a_(1)+a_(2n))+(a_(2)+a_(2n-1))+"......."+(a_(n)+a_(n+1))]`
`=-dn(a_(1)+a_(2n))`
`=-dn((a_(1)^(2)-a_(2n)^(2)))/((a_(1)+a_(2n)))=(-dn(a_(1)^(2)-a_(2n)^(2)))/((1-2n))[:.a_(2n)=a_(1)+(2n-1)d]`
`=(n)/(2n-1)(a_(1)^(2)-a_(2n)^(2))`.