If the 4th, 10th and 16th terms of a GP are x, y and z respectively prove that x, y, z are in GP. Class 9

If the 4^th, 10^th and 16^th terms of a GP are x, y and z respectively prove that x, y, z are in GP. Class 9

Question 1.

If the 4^th, 10^th and 16^th terms of a GP are x, y and z respectively prove that x, y, z are in GP. Class 9

Solution:

Given, t₄ = x, t₁₀ = y and t₁₆ = z

Using t_n = ar^n-1

t₄ = ar³

∴ x = ar³

t₁₀ = ar⁹

∴ y = ar⁹

t₁₆ = ar¹⁵

∴ z = ar¹⁵

Now, to prove x,y, z are in GP, ratio between the consecutive terms should be the same.

$\begin{aligned} & \frac{t_2}{t_1}=\frac{y}{x}=\frac{a r^9}{a r^3}=r^6 \\ & \frac{t_3}{t_2}=\frac{z}{y}=\frac{a r^{15}}{a r^9}=r^6 \\ & \text { Since } \frac{y}{x}=\frac{z}{y}=r^6\end{aligned}$

∴ x, y, z are in GP

Hence Proved.