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CBSE Class 10 Answered

If the sum of the first π‘š terms of an AP is 𝑛 and the sum of first 𝑛 terms is π‘š. Then, show that the sum of its first (π‘š + 𝑛) terms is −(π‘š + 𝑛).
Asked by yosuraj1234 | 05 Mar, 2019, 03:16: PM
answered-by-expert Expert Answer
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)
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