Question

If $m$ arithmetic means $(A.Ms)$ and three geometric means $(G.Ms)$ are inserted between $3$ and $243$ such that $4^{th}$ $A.M.$ is equal to $2^{nd}$ $G.M.$, then $m$ is equal to

Answer 1

Let $m$ arithmetic means be $A_1,A_2,A_3,A_4,\cdots ,A_m$
$3,A_1,A_2,A_3,\cdots ,A_m,243$
$\Rightarrow d=\frac{243-3}{m+1}=\frac{240}{m+1}$
$A_4=3+4\left(\frac{240}{m+1}\right)\cdots \left(1\right)$

Let $3$ geometric means be $G_1,G_2,G_3$
$3,G_1,G_2,G_3,243$
$\Rightarrow r={\left(\frac{243}{3}\right)}^{\frac{1}{3+1}}=(81{)}^{\frac{1}{4}}=3$
$G_2=a{r}^2=3\times 3^2=27\cdots \left(2\right)$

Equating $\left(1\right)$ and $\left(2\right)$, we get
$\Rightarrow 27=3+4\left(\frac{240}{m+1}\right)$
$\Rightarrow m+1=40$
$\therefore m=39$