Single correct · +4 / −1

The middle term in the expansion of (x + 1/x)^(2n) is:

  1. (a) C(2n, n)
  2. (b) C(2n, n-1)
  3. (c) 2 × C(2n, n)
  4. (d) C(2n, n) / x^n
Answer and solution

Answer (a) C(2n, n)

Solution

Maths sir · 4 steps

2n + 1 terms means one exact middle, the (n + 1)th term, and there the powers of x cancel completely.

  1. Find the middle. The power 2n is even, so the expansion has 2n + 1 terms, an odd number. There is exactly one middle term, the (n+1)th, which means r = n.
  2. General term. Write Tᵣ₊₁ = C(2n, r) x²ⁿ⁻ʳ (1/x)ʳ and combine the powers of x.

    Tᵣ₊₁ = C(2n, r) x²ⁿ⁻²ʳ

  3. Put r = n. The power becomes 2n - 2n = 0, so x⁰ = 1. The middle term is C(2n, n), a constant with no x in it.
  4. So. The middle term is C(2n, n), which is option (a).

The idea to keep. For (x + 1/x)²ⁿ, every term carries x²ⁿ⁻²ʳ. At the middle, r = n, the x's cancel, so the middle term is also the constant term.

Answer from the NTA final answer key. Solution by PrepWiser.

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