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Last updated at May 29, 2018 by Teachoo

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Misc 20 If a, b, c are in A.P, ; b, c, d are in G.P and 1/c, 1/d, 1/e are in A.P. prove that a, c, e are in G.P. It is given that a, b, c are in AP So, their common difference is same b a = c b b + b = c + a 2b = c + a b = ( + )/2 Also given that b, c, d are in GP So, their common ratio is same / = / c2 = bd Also 1/c, 1/d, 1/e are in A.P. So, their common difference is same 1/d 1/c= 1/e 1/d 1/d + 1/d = 1/e + 1/c 2(1/d) = (c + e)/ec 2/d = (c + e)/ec d/2 = ec/(c + e) d = 2(ec/(c+e)) We need to show that a, c, e are in GP i.e. we need to show their common ratio is same c/a = e/c c2 = ae So, we need to show c2 = ae From (2), we have c2 = bd Putting value of b = ( + )/2 & d = 2 /( + ) c2 = ((a + c)/2)(2ce/(c + e)) c2 = ((a + c)(2ce))/(2(c + e)) c2 = ((a + c)(ce))/((c + e)) 2/ = ( ( + ))/( + ) c = ( ( + ))/( + ) c(c + e) = e(a + c) c2 + ec = ea + ec c2 = ea + ec ec c2 = ea + 0 c2 = ea Which is what we need to prove Hence proved Thus, a, c & e are in GP

Miscellaneous

Misc 1

Misc 2

Misc 3 Important

Misc 4

Misc 5

Misc 6

Misc 7 Important

Misc 8

Misc 9

Misc 10

Misc 11

Misc 12

Misc 13

Misc 14

Misc 15

Misc 16 Important

Misc 17

Misc 18

Misc 19 Important

Misc 20 You are here

Misc 21

Misc 22

Misc 23

Misc 24

Misc 25 Important

Misc 26

Misc 27

Misc 28 Important

Misc 29

Misc 30

Misc 31

Misc 32 Important

Chapter 9 Class 11 Sequences and Series

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About the Author

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.