Last updated at Dec. 8, 2016 by Teachoo

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Example 22 If a, b, c are in G.P. and "a" ^(1/๐ฅ) = "b" ^(1/๐ฆ) = "c" ^(1/๐ง) , prove that x, y, z are in A.P. Given that "a" ^(1/๐ฅ) = "b" ^(1/๐ฆ) = "c" ^(1/๐ง) Let "a" ^(1/๐ฅ) = "b" ^(1/๐ฆ) = "c" ^(1/๐ง) = k Now, "a" ^(1/๐ฅ) = k Taking power x both sides ("a" ^(1/๐ฅ) )^๐ฅ = ใ"(k)" ใ^๐ฅ "a" ^(๐ฅ ร 1/๐ฅ) = "k" ^๐ฅ a = "k" ^๐ฅ Also, "b" ^(1/๐ฆ) = k Taking power y both sides ("b" ^(1/๐ฆ) )^๐ฆ = ใ"(k)" ใ^๐ฆ "b" ^(๐ฆ ร 1/๐ฆ) = "k" ^๐ฆ b= "k" ^๐ฆ Similarly, "c" ^(1/๐ง) = k Taking power z both sides ("c" ^(1/๐ง) )^๐ง = ใ"(k)" ใ^๐ง "c" ^(๐ง ร 1/๐ง) = "k" ^๐ง c = "k" ^๐ง Thus, a = "k" ^๐ฅ , b = "k" ^๐ฆ & c = "k" ^๐ง It is given that a, b & c are in GP So, ratio will be the same ๐/๐ = ๐/๐ b2 = ac putting value of a, b & c from (1) ("k" ^๐ฆ )^2 = "k" ^๐ฅ "k" ^๐ง "k" ^2๐ฆ = "k" ^(๐ฅ+๐ง) Comparing powers 2y = x + z We need to show x, y & z are in AP i.e. we need to show that their common difference is same i.e. we need to show y โ x = z โ y y + y = z + x 2y = z + x And we have proved in (2) that 2y = z + x Hence, x, y & z are in A.P. Hence proved

Examples

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Example 2

Example 3

Example 4

Example 5

Example 6 Important

Example 7

Example 8

Example 9

Example 10

Example 11

Example 12

Example 13

Example 14 Important

Example 15 Important

Example 16

Example 17 Important

Example 18 Important

Example 19 Important

Example 20

Example 21 Important

Example 22 You are here

Example 23 Important

Example 24 Important

Chapter 9 Class 11 Sequences and Series

Serial order wise

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.