Let a₁,a₂ ,...,a_n be sequence of real numbers, with ${\mathrm{a}}_{\mathrm{n}+1}={\mathrm{a}}_{\mathrm{n}}+\sqrt{1+{\mathrm{a}}_{\mathrm{n}}^2}$ and a₀ = 0 if $\underset{\mathrm{n}\to \mathrm{\infty }}{lim} \frac{{\mathrm{a}}_{\mathrm{n}}}{2^{\mathrm{n}-1}} \mathrm{is} \frac{\mathrm{k}}{\mathrm{\pi }}$, then k is equal to

Let ${\mathrm{a}}_{\mathrm{n}}=\cot {\mathrm{\alpha }}_{\mathrm{n}}\Rightarrow {\mathrm{a}}_{\mathrm{n}+1}=\cot {\mathrm{\alpha }}_{\mathrm{n}}+\cos {\mathrm{ec\alpha }}_{\mathrm{n}}$
$=\frac{\cos \left({\mathrm{\alpha }}_{\mathrm{n}}\right)}{\sin \left({\mathrm{\alpha }}_{\mathrm{n}}\right)}+\frac{1}{\sin \left({\mathrm{\alpha }}_{\mathrm{n}}\right)}=\frac{2{\cos }^2\left(\frac{{\mathrm{\alpha }}_{\mathrm{n}}}{2}\right)}{2\sin \left(\frac{{\mathrm{\alpha }}_{\mathrm{n}}}{2}\right)\cos \left(\frac{{\mathrm{\alpha }}_{\mathrm{n}}}{2}\right)}=\cot \left(\frac{{\mathrm{\alpha }}_{\mathrm{n}}}{2}\right)$
also ${\mathrm{a}}_0=0\Rightarrow {\mathrm{a}}_1=1\Rightarrow \cot \left({\mathrm{\alpha }}_1\right)=1\Rightarrow {\mathrm{\alpha }}_1=\frac{\mathrm{\pi }}{4}$
$\Rightarrow {\mathrm{a}}_2=\cot \left(\frac{\mathrm{\pi }}{4.2}\right),{\mathrm{a}}_3=\cot \left(\frac{\mathrm{\pi }}{4{.2}^2}\right),\dots ,{\mathrm{a}}_{\mathrm{n}}=\cot \left(\frac{\mathrm{\pi }}{4{.2}^{\mathrm{n}-1}}\right)$
Put $\frac{1}{2^{\mathrm{n}-1}}=\mathrm{x}$
Hence $\underset{\mathrm{n}\to \mathrm{\infty }}{lim} \left(\frac{{\mathrm{a}}_{\mathrm{n}}}{2^{\mathrm{n}-1}}\right)=\underset{\mathrm{n}\to \mathrm{\infty }}{lim} \frac{1}{\left(\frac{\tan \left(\frac{\mathrm{\pi x}}{4}\right)}{\mathrm{n}}\right)}=\frac{4}{\mathrm{\pi }}$