Question

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

Answer 1

The 4 th , 10 th and 16 th terms of a G.P. are x, yand z respectively.

The formula for n th term of a G.P. is given by,

a n =a r n−1 (1)

Substitute the values in equation (1) to obtain the 4 th term.

a 4 =a r 4−1 x=a r 3 (2)

Substitute the values in equation (1) to obtain the 10 th term.

a 10 =a r 10−1 y=a r 9 (3)

Substitute the values in equation (1) to obtain the 16 th term.

a 16 =a r 16−1 z=a r 15 (4)

Divide equation (3) by equation (2).

y x = a r 9 a r 3 y x = r 6 (5)

Divide equation (4) by equation (3).

z y = a r 15 a r 9 z y = r 6 (6)

Equate the terms of equation (5) and equation (6).

y x = z y y 2 =zx y= zx

Here y is the geometric mean of x and z.

Thus, x, yand z are in a G.P.