Question

If in an AP the sum of first m terms= n and the sum of first n terms= m, prove that sum of (m+n) term is -(m+n).

Answer 1

First term = a , common difference = d

Sum ofmterms= n

→m/2(2a + (m-1)d) = n …….(1)

Sum of nterms= m

→n/2(2a + (n-1)d) = m ……..(2)

Eq. (2) – (1)

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

n-m{2a + d(n+m-1)}=-2(n-m) …….(3)

2a + d(n+m-1) = -2

Now, Sum ofm+nterms= m+n/2{2a+(m+n-1)d}

=-(m+n) (using eq. 3)