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

$$\begin{gathered} a_4=x \\ \Rightarrow a{r}^3=x \\ \mathrm{Also},a_{10}=y \\ \Rightarrow a{r}^9=y \\ \mathrm{And},a_{16}=z \\ \Rightarrow a{r}^{15}=z \\ \because \frac{y}{x}=\frac{a{r}^9}{a{r}^3}=r^6 \\ \mathrm{and}\frac{z}{y}=\frac{a{r}^{15}}{a{r}^9}=r^6 \\ \therefore \frac{y}{x}=\frac{z}{y} \\ \mathrm{Therefore},x,y\mathrm{and}z\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \end{gathered}$$