The coefficient of the middle term in (1 + x)^(2n) is twice the coefficient of the preceding term.

Solution

Maths sir · 4 steps

Divide the two neighbouring coefficients: the ratio is (n+1)/n, which equals 2 only at n = 1.

1. Find the middle term. (1+x)²ⁿ has 2n+1 terms, an odd number, so there is exactly one middle term. It is Tₙ₊₁ = C(2n, n)xⁿ, and the term before it is Tₙ = C(2n, n-1)xⁿ⁻¹.
2. Take the ratio. Use the standard result C(m, r)/C(m, r-1) = (m-r+1)/r with m = 2n and r = n.

   C(2n, n)/C(2n, n-1) = (2n-n+1)/n = (n+1)/n

3. Check against 2. (n+1)/n = 2 gives n = 1 only. For n = 2 the coefficients are 1, 4, 6, 4, 1, and 6 is not twice 4.
4. So. The ratio is (n+1)/n, not 2, for every n ≥ 2. The statement is false.

The idea to keep. To compare neighbouring binomial coefficients, use C(m, r)/C(m, r-1) = (m-r+1)/r. It saves you from expanding factorials every time.

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