Given $f[n]=a^{n}u[n],$ find the $z-$transforms of the following:

$($a$)f[\frac{n}{7}]$

$($b$)f[n-7]u[n-7]$; verify your result by a power series expansion

$($c$)f[n+3]u[n]$; verify your result by a power series expansion

$($d$)b^{2n}f[n]$; using two different procedures.