If $m$ arithmetic means (A.M's) and three geometric means (G.M's) are inserted between 3 and 243 suc

Question

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

Professor PC Thomas Classes · Answer

(39) : Given, $3, A_{1}, A_{2}, \ldots A_{m}, 243$\ $	herefore \quad r=\frac{243-3}{m+1}=\frac{240}{m+1}$\ Also, $3, G_{1}, G_{2}, G_{3}, 243$ $$\begin{equation*}r=\left(\frac{243}{3}ight)^{\frac{1}{3+1}}=(81)^{\frac{1}{4}}=3 	ag{i} \end{equation*}$$ Now, $G_{2}=A_{4}$\ $\Rightarrow a r^{2}=a+4 d \Rightarrow 3(3)^{2}=3+4\left(\frac{240}{m+1}ight)$ $$\begin{equation*}\Rightarrow \quad 27=3+\frac{960}{m+1} \Rightarrow m+1=40 \Rightarrow m=39 	ag{ii} \end{equation*}$$