Part of ALG-06 — Binomial Theorem

Middle Term

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For (x + y)^n:

  • If n is even: One middle term = Tn/2+1T_{n/2 + 1}
  • If n is odd: Two middle terms = T_{n+12\frac{n+1}{2}} and T_{n+32\frac{n+3}{2}}

The middle term typically has the largest absolute value when x and y have comparable magnitudes, and always involves the largest binomial coefficient C(n, floorn2\frac{n}{2}).

Example: In (1 + x)^6, middle term is T4T_4 = C(6,3)x3x^3 = 20x3x^3. In (1 + x)^7, middle terms are T4T_4 = C(7,3)x3x^3 = 35x3x^3 and T5T_5 = C(7,4)*x4x^4 = 35x4x^4.

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