Check sibling questions

Ex 9.3, 22 - If pth, qth and rth terms of a GP are a, b, c - Geometric Progression(GP): Calculation based/Proofs

Ex 9.3, 22 - Chapter 9 Class 11 Sequences and Series - Part 2
Ex 9.3, 22 - Chapter 9 Class 11 Sequences and Series - Part 3 Ex 9.3, 22 - Chapter 9 Class 11 Sequences and Series - Part 4

This video is only available for Teachoo black users

Get Real time Doubt solving from 8pm to 12 am!


Transcript

Ex9.3, 22 If the pth ,qth and rth terms of a G.P. are a, b and c, respectively. Prove that aq r br p cp q = 1 We know that nth term of G.P = ARn 1 (We are using a, r in the question, so we use A for first term and R for common ratio) It is given that pth term of G.P = a Ap = a ARp 1 = a a = ARp 1 aq r = ("ARp 1")q r We need to show that aq r br p cp q = 1 Also, qth term of G.P = b Aq = b ARq 1 = b b = ARq 1 br p = (ARq 1)r p & rth term of G.P = c Ar = c ARr 1 = c c = ARr 1 cp q = (ARr 1)p q Now, our equations are aq r = ("ARp 1")q r (1) br p = (ARq 1)r p (2) & cp q = (ARr 1)p q (3) Taking L.H.S. aq r br p cp q Putting values from (1), (2) & (3) = ("ARp 1")q r (ARq 1)r p (ARr 1)p q = "Aq r R(p 1")(q r) "Ar p R(q 1")(r p) "Ap q R(r 1")(p q) = "Aq r Ar p Ap q " "R(p 1")(q r) "R(q 1")(r p) "R(r 1")(p q) = A(q r) + (r p) + (p q) R(p 1) (q - r) + (q 1) ( r - p) + (r 1) ( p-q) = A(q r) + (r p) + (p q) Rp(q r ) 1(q r) + q( r p) 1(r p) +r(p q) 1( p q) = A(q q) + (r r) + (p p) Rpq pr q + r + qr qp r + p + rp qr p + q = A0 R pq pq q + q + r r + p p + qr qr + qp qr = A0 R0 = 1 1 = 1 = R.H.S Thus, L.H.S = R.H.S Hence proved

Davneet Singh's photo - Co-founder, Teachoo

Made by

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.