The binomial theorem: expansion, general term and sums of expansions

The expansions of (x + a)ⁿ and (x − a)ⁿ differ only in signs. The terms with even r (T1, T3, T5, …) are the same in both; the terms with odd r (T2, T4, …) change sign. So adding cancels the odd-r terms and doubles the others: (x + a)ⁿ + (x − a)ⁿ = 2(T1 + T3 + T5 + …). Subtracting does the opposite: (x + a)ⁿ − (x − a)ⁿ = 2(T2 + T4 + …), where the T’s are the terms of (x + a)ⁿ. The table shows it for n = 5. Sometimes a long sum is secretly one expansion. If it reads Aⁿ + nC1 Aⁿ⁻¹B + nC2 Aⁿ⁻²B² + … + Bⁿ, it equals (A + B)ⁿ; if the signs alternate it equals (A − B)ⁿ. Find A and B by looking at the powers that fall in one bracket and rise in the other. Finally, when a question gives a coefficient or a ratio of coefficients, turn each into an equation with nCr. Two pieces of information give two unknowns; dividing one equation by another often cancels what you do not need.