Middle Term

Question

When the expression (x + 1/x)^{10} is fully expanded, the single middle term of the resulting expansion is best described by which of the following?

RankGuru · Accepted Answer

For a binomial raised to an even power n, the expansion has n + 1 terms and exactly one middle term, namely the (n/2 + 1)th term, a standard JEE Advanced fact. Here n = 10, so there are 11 terms and the single middle term is the 6th term, T_6, corresponding to r = 5 in T_{r+1} = C(n, r) a^{n-r} b^r. With a = x and b = 1/x, we get T_6 = C(10, 5) x^{10-5} (1/x)^5 = C(10, 5) x^5 x^{-5} = C(10, 5) x^0 = 252. Thus the middle term is the pure constant 252. Option 210 x^2 corresponds to a non-middle term and retains a power of x, so it cannot be the central one. Option 120 = C(10, 3) is the wrong coefficient from r = 3. Option 252 x mistakenly attaches a power of x even though the exponents cancel. The central binomial coefficient C(10,5) = 252 is the largest in the row, consistent with the symmetry of Pascal's triangle. Plausibility check: since a = x and b = 1/x are reciprocals, the middle term must be dimensionless, and indeed the net exponent 5 - 5 = 0 confirms a constant.

Middle Term

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