Examples
Example 1 (ii)
Example 2
Example 3 Important
Example 4 Deleted for CBSE Board 2023 Exams
Example 5 Deleted for CBSE Board 2023 Exams
Example 6 Important Deleted for CBSE Board 2023 Exams
Example 7 Deleted for CBSE Board 2023 Exams
Example 8 Important
Example 9
Example 10 Important
Example 11
Example 12 Important
Example 13
Example 14 Important
Example 15 Important
Example 16
Example 17 Important
Example 18 Important
Example 19 Important Deleted for CBSE Board 2023 Exams
Example 20 Deleted for CBSE Board 2023 Exams
Example 21 Important
Example 22 You are here
Example 23
Example 24 Important
Examples
Last updated at Dec. 8, 2016 by Teachoo
Example 22 If a, b, c are in G.P. and "a" ^(1/𝑥) = "b" ^(1/𝑦) = "c" ^(1/𝑧) , prove that x, y, z are in A.P. Given that "a" ^(1/𝑥) = "b" ^(1/𝑦) = "c" ^(1/𝑧) Let "a" ^(1/𝑥) = "b" ^(1/𝑦) = "c" ^(1/𝑧) = k Now, "a" ^(1/𝑥) = k Taking power x both sides ("a" ^(1/𝑥) )^𝑥 = 〖"(k)" 〗^𝑥 "a" ^(𝑥 × 1/𝑥) = "k" ^𝑥 a = "k" ^𝑥 Also, "b" ^(1/𝑦) = k Taking power y both sides ("b" ^(1/𝑦) )^𝑦 = 〖"(k)" 〗^𝑦 "b" ^(𝑦 × 1/𝑦) = "k" ^𝑦 b= "k" ^𝑦 Similarly, "c" ^(1/𝑧) = k Taking power z both sides ("c" ^(1/𝑧) )^𝑧 = 〖"(k)" 〗^𝑧 "c" ^(𝑧 × 1/𝑧) = "k" ^𝑧 c = "k" ^𝑧 Thus, a = "k" ^𝑥 , b = "k" ^𝑦 & c = "k" ^𝑧 It is given that a, b & c are in GP So, ratio will be the same 𝑏/𝑎 = 𝑐/𝑏 b2 = ac putting value of a, b & c from (1) ("k" ^𝑦 )^2 = "k" ^𝑥 "k" ^𝑧 "k" ^2𝑦 = "k" ^(𝑥+𝑧) 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 – x = z – 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