Hello purebladeknightYou'll have realised, of course, that the 1st, 3rd, ... terms (the odd terms) in the expansions of and have been eliminated by subtraction, leaving us with the even terms, which are then added together to give the factor of in each term in the final expression.

Then notice that in the expansion of there are terms altogether, the last two terms beingSo if is odd, is even, and the final term will be .

And if is even, is odd, and the final term will be .

Grandad