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
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