Question

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.

Answer 1

$a_4=x$

$\Rightarrow a{r}^3=x$

Also, $a_{10}=y$

$\Rightarrow a{r}^9=y$

And, $a_{16}=z$

$\Rightarrow a{r}^{15}=z$

$\because \frac{y}{x}=\frac{a{r}^9}{a{r}^3}=r^6$

and $\frac{z}{y}=\frac{a{r}^{15}}{a{r}^9}=r^6$

$\therefore \frac{y}{x}=\frac{z}{y}$

Therefore, x, y and z are in G.P.