If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x,y, z are in G.P

To prove that x, y, and z are in a geometric progression (G.P.), we need to show that the ratio between any two consecutive terms is constant.

Let's assume that the common ratio of the G.P. is r.

The 4th term of the G.P. is given as x. Using the formula for the nth term of a G.P., we can write:

x = a * r^3, where a is the first term of the G.P.

Similarly, the 10th term of the G.P. is given as y:

y = a * r^9

And the 16th term of the G.P. is given as z:

z = a * r^15

To prove that x, y, and z are in a G.P., we need to show that the ratio between any two consecutive terms is constant. Let's consider the ratio between the 10th and 4th terms:

y/x = (a * r^9) / (a * r^3) = r^6

Similarly, let's consider the ratio between the 16th and 10th terms:

z/y = (a * r^15) / (a * r^9) = r^6

Since both ratios are equal to r^6, we can conclude that x, y, and z are in a G.P. with a common ratio of r^6.

Therefore, we have proven that x, y, and z are in a G.P.