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Question 28 - CBSE Class 10 Sample Paper for 2020 Boards - Maths Standard - Solutions of Sample Papers for Class 10 Boards

Last updated at Dec. 16, 2024 by Teachoo

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.

 

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/

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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo