$2\mathrm{.}15$ The $n-$bit two's complement integer $N$ is written $a_{n-1},a_{n-2},dots{a}_1,a_0.$ Prove that $($in two's complement notation$)$ the representation of a signed binary number in $n+1$ bits may be derived from its representation in $n$ bits by repeating the leftmost bit. For example, if $n=-12=10100$ in five bits or $n=-12=110100$ in six bits.