Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Question:

Insert two numbers between 3 and 81 so that the resulting sequence is G.P.

Solution:

Let $G_{1}$ and $G_{2}$ be two numbers between 3 and 81 such that the series, $3, G_{1}, G_{2}, 81$, forms a G.P.

Let $a$ be the first term and $r$ be the common ratio of the G.P.

$\therefore 81=(3)(r)^{3}$

$\Rightarrow r^{3}=27$

$\therefore r=3$ (Taking real roots only)

For $r=3$,

$G_{1}=a r=(3)(3)=9$

$G_{2}=a r^{2}=(3)(3)^{2}=27$

Thus, the required two numbers are 9 and 27 .