Between two numbers whose sum is 2$\frac{1}{6}$, an even  number of arithmetic means are inserted. If the sum of these means exceeds their number by unity, then the number of means are

Let 2n arithmetic means be A1, A2, A3, …, A2n between a and b. 

$\mathrm{Then} {\mathrm{A}}_1+{\mathrm{A}}_2+{\mathrm{A}}_3+\dots +{\mathrm{A}}_{2\mathrm{n}}=\frac{\mathrm{a}+\mathrm{b}}{2}\times 2\mathrm{n}$

$=\frac{\frac{13}{6}}{2}\times 2\mathrm{n}=\frac{13\mathrm{n}}{6}$

Given, ${\mathrm{A}}_1+{\mathrm{A}}_2+{\mathrm{A}}_3+\dots +{\mathrm{A}}_{2\mathrm{n}}=2\mathrm{n}+1$

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}2\mathrm{n}+1=\frac{13\mathrm{n}}{6};\text{ or }12\mathrm{n}+6=13\mathrm{n}\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{n}=6\end{array}$

$\therefore \text{ The number of means }=2\mathrm{n}=2\times 6=12\text{. }$