Middle terms, sums of coefficients and the term containing a given power of x

The expansion of (x + a)ⁿ has n + 1 terms. If n is even, n + 1 is odd, so there is exactly one middle term, number n/2 + 1 from the start. If n is odd, n + 1 is even, so there are two middle terms, number (n + 1)/2 and number (n + 3)/2. For n = 8 there are 9 terms and the middle one is T5 (r = 4); for n = 7 there are 8 terms and the middle ones are T4 and T5 (r = 3 and r = 4). Once you know the order, use the general term of Week 1. When you add or subtract two expansions, first count the terms that survive: (x + a)ⁿ + (x − a)ⁿ keeps only T1, T3, T5, …, and (x + a)ⁿ − (x − a)ⁿ keeps only T2, T4, …, so there are about half as many terms and the middle term is the middle one of that shorter list. The middle terms are where the numbers nCr are largest, because Pascal’s rows rise to the centre and then fall.