Question

Insert five geometric means between $\frac{1}{3}$ and $9$. If their product is the nth power of the single geometric mean between them find $n$.

Answer 1

5 G.M. is between $\frac{1}{3}\&9\frac{1}{3}{r}^{7-1}=9r^6=27r=\sqrt{3}$

Product of all G.M.'s $=a^5{r}^{15}=\frac{3^{\frac{15}{2}}}{3^5}=3^{\frac{5}{2}}$

1 G.M. between $\frac{1}{3}\&9\frac{1}{3}.r^2=9r=3^{\frac{3}{2}}$

Inserted G.M. $=\frac{3^{\frac{3}{2}}}{3}=3^{\frac{1}{2}}$

Given $(3^{\frac{1}{2}}{)}^n=3^{\frac{5}{2}}n=5$