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If a(m) be the mth term of an AP, show t...

If `a_(m)` be the mth term of an AP, show 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))`.

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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))`.
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