Insert two numbers between $3$ and $81$ so that the resulting sequence is $G.P.$
Insert two numbers between $3$ and $81$ so that the resulting sequence is $G.P.$
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 $G.P.$
$\therefore 81=(3)(r)^{3}$
$\Rightarrow r^{3}=27$
$\therefore r=3$ (Talking 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 $27$
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