Binomial Theorem and Sequences
Binomial Theorem and Sequences tests expansion, general and middle terms, coefficients, and AP/GP formulas. The exam often combines binomial coefficients with sequences or asks for a particular term without full expansion.
Core rules
(a+b)ⁿ=Σ_{r=0}^{n} nCr a^{n−r}bʳ. The general term is T_{r+1}=nCr a^{n−r}bʳ; note the index shift. If n is even, there is one middle term, T_{n/2+1}; if n is odd, there are two middle terms. For an AP, nth term a+(n−1)d and sum Sₙ=n/2[2a+(n−1)d]. For a GP, nth term arⁿ⁻¹, finite sum a(rⁿ−1)/(r−1) for r≠1, and infinite sum a/(1−r) for |r|<1. AM between a and b is (a+b)/2; GM is √(ab).
Worked example
Find the middle term in (2x+3)⁶. Here n=6 is even, so the middle term is T₄, with r=3. T₄=6C3(2x)³(3)³=20·8x³·27=4320x³.
Traps
T_{r+1} is not T_r; check the index. In (a−b)ⁿ, the sign alternates, so include (−1)ʳ. The middle term is not always the coefficient of xⁿ. For an infinite GP, the formula requires |r|<1. Do not confuse nth term with sum to n terms.