Let the positive numbers ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,{\mathrm{a}}_4\mathrm{and}{\mathrm{a}}_5$ be in a G.P. Let their mean and variance be $\frac{31}{10}$ and $\frac{\mathrm{m}}{\mathrm{n}}$ respectively, where m and n are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and ${\mathrm{a}}_3+{\mathrm{a}}_4+{\mathrm{a}}_5=14,$ then m + n is equal to_________.

$$\begin{gathered} \mathrm{a}(\frac{1}{{\mathrm{r}}^2}+\frac{1}{\mathrm{r}}+1+\mathrm{r}+{\mathrm{r}}^2)=\frac{31}{2} \\ \frac{1}{\mathrm{a}}(\frac{1}{{\mathrm{r}}^2}+\frac{1}{\mathrm{r}}+1+\mathrm{r}+{\mathrm{r}}^2)=\frac{31}{8} \\ \Rightarrow {\mathrm{a}}^2=4\Rightarrow \mathrm{a}=2 \\ \mathrm{again} \frac{1}{{\mathrm{r}}^2}+\frac{1}{\mathrm{r}}+1+\mathrm{r}+{\mathrm{r}}^2=\frac{31}{4} \\ \Rightarrow {(\mathrm{r}+\frac{1}{\mathrm{r}})}^2+(\mathrm{r}+\frac{1}{\mathrm{r}})=\frac{31}{4}+1=\frac{35}{4} \\ \mathrm{Put} \mathrm{r}+\frac{1}{\mathrm{r}}=\mathrm{t} \mathrm{then}, \\ 4{\mathrm{t}}^2+4\mathrm{t}-35=0 \\ \Rightarrow \mathrm{t}=5/2\Rightarrow \mathrm{r}=2 \end{gathered}$$
$\therefore$Numbers are $\frac{1}{2},\hspace{0.17em}\hspace{0.17em}1,\hspace{0.17em}\hspace{0.17em}2,\hspace{0.17em}\hspace{0.17em}4,\mathrm{\hspace{0.17em}}8\mathrm{\hspace{0.17em}}$
$$\begin{gathered} {\mathrm{\sigma }}^2=\frac{\frac{1}{4}+1+4+16+64}{5}-{\left(\frac{31}{10}\right)}^2=\frac{341}{20}-\frac{961}{100} \\ \Rightarrow {\mathrm{\sigma }}^2=\frac{186}{25}=\frac{\mathrm{m}}{\mathrm{n}} \\ \therefore \mathrm{m}+\mathrm{n}=211 \end{gathered}$$