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Question

If in an AP the sum of m terms is equal to n end the sum of n terms is equal to m then prove that the sum of (m + n) terms is -(m+n)


Solution

First term = a , common difference = d

Sm = n

→m/2(2a  + (m-1)d) = n  …….(1)

Sn = 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, Sm+n = m+n/2{2a+(m+n-1)d}

=-(m+n)        (using eq. 3)

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