A sequence of real number a₁, a₂, a₃......a_n+1 is such that ${\mathrm{a}}_1=0,\left|{\mathrm{a}}_2\right|=\left|{\mathrm{a}}_1+1\right|$ $\left|{\mathrm{a}}_3\right|=\left|{\mathrm{a}}_2+1\right|,\dots \dots .\left|{\mathrm{a}}_{\mathrm{n}+1}\right|=\left|{\mathrm{a}}_{\mathrm{n}}+1\right|$then $\frac{{\mathrm{a}}_1+{\mathrm{a}}_2+\dots \dots {\mathrm{a}}_{\mathrm{n}}}{\mathrm{n}}$ cannot be equal to

$\begin{array}{l}{\mathrm{a}}_2^2={\mathrm{a}}_1^2+2{\mathrm{a}}_1=1 , {\mathrm{a}}_3^2={\mathrm{a}}_2^2+2{\mathrm{a}}_2=1 ····· {\mathrm{a}}_{\mathrm{n}+1}^2={\mathrm{a}}_{\mathrm{n}}^2+2{\mathrm{a}}_{\mathrm{n}+1} \\ \text{adding above all then,}\\ {\mathrm{a}}_{\mathrm{n}+1}^2={\mathrm{a}}_1^2+2\left({\mathrm{a}}_1+{\mathrm{a}}_2+\dots \dots +{\mathrm{a}}_{\mathrm{n}}\right)+\mathrm{n}\\ \therefore {\mathrm{a}}_1+{\mathrm{a}}_2+{\mathrm{a}}_3+\dots ..+{\mathrm{a}}_{\mathrm{n}}\\ =\frac{{\mathrm{a}}_{\mathrm{n}+1}^2-1}{2.\mathrm{n}}\\ =-\frac{1}{2}+\frac{{\mathrm{a}}_{\mathrm{n}+1}^2}{2\mathrm{n}}-\frac{1}{2}\end{array}$