Problem $1$: $[10$ points$]$

$($a$)$ Prove that for a given pressure ratio $($po$2/$po$1)$ and technology level $($e$.$g$.$ polytropic efficiency

$\backslash$eta$\_$p$),$ compressor power is proportional to compressor$-$inlet temperature

$($b$)$ Prove that for a given pressure ratio $($po$5/$po$4)$ and technology level $($e$.$g$.$ polytropic efficiency

$\backslash$eta$\_$t$),$ turbine power is proportional to turbine inlet temperature

$($c$)$ What are the implications of $($a$)$ and $($b$)$ for gas turbine cycles?