What is the actual operational difference between single phase and three phase motor.

I’m doing a induction proof and need to find a way to prove (k(k+1)(k+2))/3 + k + 1 = ((k+1)(k+2)(k+3))/3. thanks

When the direction of a 3 phase induction motor is changed , is there may be any losses increases?

I know the advantages of 3phase induction motor (IM) when compared with a single-phase IM one. However I need to justify using a single-phase IM (main and auxiliary windings) with a 3 legs inverter instead a 3phase IM with also a 3 legs inverter. Are the single-phase IM cheaper the 3phase one...