Question

The two geometric means between the numbers $1$ and $64$ are
(a) $1$ and $64$
(b) $4$ and $16$
(c) $2$ and $16$
(d) $8$ and $16$
(e) $3$ and $16$

Answer 1

(b) $4$ and $16$
Let the two G.M.s between $1$ and $64$ be $G_1$ and $G_2$.

Thus, $1,G_1,G_2$ and $64$ are in G.P.

here, $a=1$

$64=a{r}^3$

$\Rightarrow r^3=4^3$

$\Rightarrow r=3$

Now,

$G_1=ar=1\left(4\right)=4$

$G_2=a{r}^2=1(4{)}^2=16$

Thus, $4$ and $16$ are the required G.M.s.