Q1 of 8 Page 180

If in an A.P the sum of m terms is equal to n and the sum of n terms is equal to m then prove that the sum of (m – n) terms is – (m + n).

Sum of m terms of AP = n



⇒ 2am + m2d – md = 2n ..(i)


Sum of n terms of the same AP = m



⇒ 2an + n2d – nd = 2m ..(ii)


Subtracting equation (ii) from (i) , I.e. (i) –(ii)


We get,


2a(m – n) + d(m2 – n2) – d(m – n) = 2 (n – m)


(m – n)[2a + d(m + n – 1)] = – 2(m – n)


⇒ (2a + (m + n – 1)d) = – 2 ..(iii)



Substituting values from equation (iii)


We get,


= – (m + n)


So, the sum of (m + n) terms is –(m + n)


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