If x, y, z are three positive numbers forming a geometric sequence, then show that `log_a x, log_a y,log_a z` form an arithmetic sequence , a being positive and not equal to 1.

If x is a positive real number different from 1 such that log_a x, log_b x, log_c x are in A.P then

If the mth, nth and pth terms of a G.P. form three consecutive terms of a geometric sequence, prove that m, n and p form three consecutive terms of an arithmetic sequence.

If x,y and z are three real numbers such that x+y+z=4 and x^2+y^2+z^2=6 ,then show that each of x,y and z lie in the closed interval [2/3,2]

If n> 1 and log_2 x,log_3 x, log_x 16 , are in G.P., then what is x equal to ?

(2x - y + z)^2 in the expanded form is :

There are four numbers such that the first three of them form an arithmetic Sequence and the last three form a geometric Sequence. The sum of the first and third terms is 2 and that of second and fourth is 26. What are these numbers ?

If x,y,z are positive real numbers, prove that: sqrt(x^-1 y).sqrt(y^-1z).sqrt(z^(-1)x)=1

Three distinct numbers x,y,z form a GP in that order and the numbers 7x+5y,7y+5z,7z+5x form an AP in that order. The common ratio of GP is

If x,y,z be three positive prime numbers. The progression in which sqrt(x),sqrt(y),sqrt(z) can be three terms (not necessarily consecutive) is

The number of three digit numbers of the form xyz such that x lt y , z le y and x ne0 , is