prove that series combination resistance have maximum and parallel combination resistance have minimum current

When two res are connected in series:

eq res:R=r1+r2

current in the ckt:   Is=V/(r1+r2)........(1)

now R ie (r1+r2)sh always be greater than individual res r1 or r2.

so the current will obviously be lesser in comparison with the situation when single res would have used.

 

in parallel combination:

eq res: R=r1r2/(r1+r2)  so R is less than individual res r1 or r2....so the value of current in paallel will be greater.

and current in the ckt: Ip=V(r1+r2)/r1r2........(2)

 

so Ip/Is  =(r1+r2)²/r1r2 sh always be greater than 1, ie current in parallel is greater than current in series.