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

This lesson ties the first two together. First, the sum of the coefficients. A coefficient is what is left of a term when every letter is replaced by 1, so putting x = 1 adds all of them. For (2x² + 1/x)⁵ the sum is (2 + 1)⁵ = 243. Putting x = −1 gives the alternating sum: the coefficients taken with their signs flipped in the odd places. For (1 + x)ⁿ, x = 1 gives nC0 + nC1 + nC2 + … = 2ⁿ, and x = −1 gives nC0 − nC1 + nC2 − … = 0 (for n ≥ 1). Adding the two results and halving, then subtracting and halving, shows nC0 + nC2 + nC4 + … = nC1 + nC3 + nC5 + … = 2ⁿ⁻¹. Second, equal terms. Two neighbouring terms are equal when their quotient T(r+2)/T(r+1) is 1. For the two middle terms of (px + qy)ⁿ, with n odd, the two numbers nCr are equal, so the terms are equal exactly when px = qy. Third, a condition on a middle term: write the middle term, equate it to the number you are given, and solve for the unknown.