Please see the attached photo. The results comes from simplifying the first principles, note that we can usually find patterns after we simplify the long equations, which means it would not be necessary to derive everything during a test or exam (that’s too time consuming). However, just be cautious with changes in Makehams, because we cannot simply apply one trick to all the problems. So its important to understand the derivation instead of only learning the trick.
The explanation is quite long but the logic should follow. Hope it helps :)