Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of (∜2+1/∜3)ⁿ, in the increasing powers of 1/∜3 be ∜6: 1. If the sixth term from the beginning is α/∜3, then α is equal to _________.

Solution

Maths sir · 5 steps

Divide the two fifth terms: the binomial coefficients cancel and only a power of 6 is left.

1. Name the terms. Let a = 2^(1/4) and b = 3^(-1/4). The general term is Tᵣ₊₁ = C(n, r) aⁿ⁻ʳ bʳ, written in increasing powers of b.
2. Fifth from each end. The fifth term from the beginning is C(n, 4) aⁿ⁻⁴ b⁴. The fifth term from the end is the fifth term of (b + a)ⁿ, which is C(n, 4) a⁴ bⁿ⁻⁴.

   (C(n, 4) aⁿ⁻⁴ b⁴)/(C(n, 4) a⁴ bⁿ⁻⁴) = (a/b)ⁿ⁻⁸ = (2^(1/4) 3^(1/4))ⁿ⁻⁸ = 6^((n-8)/4)

3. Find n. Set 6^((n-8)/4) = 6^(1/4). Then n - 8 = 1, so n = 9.
4. Sixth term. T₆ = C(9, 5) a⁴ b⁵ = 126 × 2 × 3^(-5/4) = 252/(3 × 3^(1/4)) = 84/∜3.
5. So. Matching with α/∜3 gives α = 84.

The idea to keep. The rth term from the end of (a+b)ⁿ is the rth term of (b+a)ⁿ. In the ratio of the two, the binomial coefficients cancel and you get (a/b)ⁿ⁻²⁽ʳ⁻¹⁾.

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