If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero.

 

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Note : This is similar to Ex 9.2, 10 of NCERT – Chapter 9 Class 11

Check the answer here https://www.teachoo.com/2492/618/Ex-9.2--10---If-sum-of-first-p-terms-of-AP-is-equal-to/category/Ex-9.2/

  1. Class 10
  2. Solutions of Sample Papers for Class 10 Boards

Transcript

Question 28 If the sum of first m terms of an AP is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero. We know that Sn = n/2 ( 2a + (n – 1)d ) Where, Sn = sum of n terms of A.P. n = number of terms a = first term and d = common difference Now, Sum of first m terms = Sm Sm = m/2 [2a + (m – 1)d] Sum of first n terms = Sn Sn = n/2 [2a + (n – 1)d] It is given that Sum of first m terms = Sum of first n terms m/2 [2a + (m – 1)d] = n/2 [2a + (n – 1)d] Cancelling denominator 2 both sides m[2a + (m – 1)d] = n[2a + (n – 1)d] 2am + md(m – 1) = 2an + nd(n – 1) 2am – 2an = nd (n – 1) – md(m – 1) 2a(m – n) = d[n(n – 1) – m(m – 1)] 2a(m – n) = d[n2 – n – (m2 – m)] 2a(m – n) = d[n2 – n – m2 + m] 2a(m – n) = d[n2 – m2 + m – n] 2a(m – n) = – d [ –n2 + m2 – m + n] 2a(m – n) = – d [m2 – n2 – (m – n)] 2a(m – n) = – d [(m – n) (m + n) – (m – n)] 2a(m – n) = – d(m – n) [m + n – 1] 2a(m – n) + d(m – n) [m + n – 1] = 0 (m – n) [2a + d(m + n – 1)] = 0 Therefore, m – n = 0 m = n Not possible as m ≠ n 2a + d(m + n – 1) = 0 So, 2a + d(m + n – 1) = 0 Now, finding sum of first (m + n) terms We know that, Sum of n terms = n/2 [2a + (n – 1)d] For sum of (m + n) terms, we mut n = (m + n) Sum of (m + n) term is = (m + n)/2 [2a + (m + n – 1)d] = (m + n)/2 × 0 = 0 ∴ Sum of (m + n) term is 0 Hence proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.