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If the sum of first m terms of A P is n and sum of first n terms of the same A P is m.  Show that sum of first (m+n) terms is -(m+n).   Sent fast.
Let a and d are first term and common difference of Arithmetic progression

Given : sum of first m terms are n, Hence we have,   (m/2)[ 2a + (m-1)d ] = n   or  2am + m(m-1)d = 2n   ..................(1)
Given : sum of first n terms are m, Hence we have,  (n/2)[2a + (n-1)d ] = m   or     2an + n(n-1) d  = 2m  ...................(2)

Eqn. (1) - Eqn.(2) gives,   2a ( m - n ) + [ (m2 - n2) - (m-n) ] d = 2 (n-m) ...........................(3)

dividing eqn.(30 by (m-n) on both sides,  2a + (m+n-1)d = -2  ..........................(4)

sum of first (m+n) terms = [ (m+n)/2 ] [2a + (m+n-1)d ] = [ (m+n)/2 ] (-2) = -(m+n)  ..........................(5)

we used eqn.(4) in eqn.(5)
Answered by Thiyagarajan K | 06 Mar, 2019, 12:09: AM

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