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Ex 9.2, 5 - In AP, if pth term is 1/q , qth term is 1/p, prove - Arithmetic Progression (AP): Calculation based/Proofs

Ex 9.2, 5 - Chapter 9 Class 11 Sequences and Series - Part 2
Ex 9.2, 5 - Chapter 9 Class 11 Sequences and Series - Part 3 Ex 9.2, 5 - Chapter 9 Class 11 Sequences and Series - Part 4 Ex 9.2, 5 - Chapter 9 Class 11 Sequences and Series - Part 5 Ex 9.2, 5 - Chapter 9 Class 11 Sequences and Series - Part 6


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Ex 9.2,5 In an A.P., if pth term is 1/q and qth term is 1/p , prove that the sum of first pq terms is 1/2 (pq + 1) where p q. We know that an = a + (n 1)d Where an is nth term of AP, n is the number of terms, a be the first term & d be the common difference of the A.P. It is given that pth term is 1/ i.e. ap = 1/q a + (p 1)d = 1/q Also, qth term of A.P = 1/p i.e. aq = 1/p i.e. a + (q 1)d = 1/p Now subtracting (1) from (2) i.e. (1) (2) [a + (p 1)d] [a + (q 1)d] = 1/ 1/ a + pd d [a + qd d] = 1/ 1/ a + pd d a qd + d = 1/ 1/ a a d + d + pd qd = 1/ 1/ 0 + 0 + pd qd = 1/ 1/ d(p q) = 1/ 1/ d(p q) = ( )/ d = ( )/( ( )) d = 1/ Now finding first term i.e. a Putting d = 1/ in (1) 1/ = a + (p 1)d 1/ = a + (p 1)1/ 1/ = a + / 1/ 1/ = a + 1/ 1/ 1/ 1/ = a 1/ 0 = a 1/ 1/ = a a = 1/ Therefore, a = 1/ Thus, a = 1/ & d = 1/ We need to show sum of first pq term is 1/2(pq + 1) i.e. Spq = 1/2 (pq + 1) 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 For sum of first pq terms, Putting n = pq , a = 1/ , d = 1/ Spq = pq/2 ["2 " 1/ " + (pq 1)" 1/ ] = pq/2 [2/ " + " / 1/ ] = pq/2 [(2 + 1 )/ ] = pq/2 [(1 + )/ ] = 1/2 [pq + 1] Thus, sum of first pq terms is 1/2 (pq+1) 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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.