Question

logx a^n , logx b^n , logx c^n are in AP prove that a, b, c are in GP ​

Answer 1

Step-by-step explanation:

$\because log_x a^n, \: log_x b^n, \: log_x c^n \\ are \: in \: AP \\ \\ \therefore \: log_x b^n - log_x a^n = log_x c^n - log_x b^n \\ \\ \therefore \: log_x b^n + log_x b^n= log_x c^n + log_x a^n \\ \\ \therefore \: 2log_x b^n= log_x (c^n \times a^n )\\ \\ \therefore \: log_x b^{2n}= log_x (ac)^n \\ \\ \therefore \: b^{2n}= (ac)^n \\ \\\therefore \: b^{2}= ac \\ \\ \huge \purple{ \boxed{\therefore \: b= \sqrt{ac} }}$

Thus, a, b, c are in G. P.

Hence Proved.

Answer 2

Answer:

Please see the attachment