Let α be the constant term in the binomial expansion of (√x-6/x^(3/2))ⁿ, n ≤ 15. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x⁻ⁿ is λ α, then λ is equal to _____________.

Solution

Maths sir · 4 steps

Put x = 1 to get the sum of all coefficients, take away α, and the 649 tells you n.

1. General term. Tᵣ₊₁ = C(n, r)(√x)ⁿ⁻ʳ(-6/x^(3/2))ʳ = C(n, r)(-6)ʳ x^((n-4r)/2). The constant term needs n = 4r, so n is a multiple of 4: n = 4, 8 or 12.
2. Find n. Putting x = 1 gives the sum of all coefficients, (1-6)ⁿ = (-5)ⁿ. The remaining terms sum to (-5)ⁿ - α = 649. For n = 4: α = C(4, 1)(-6) = -24, and 625 - (-24) = 649. This works. For n = 8 or 12, (-5)ⁿ is far too large to fit.
3. Coefficient of x^(-4). We need (4-4r)/2 = -4, so r = 3. The coefficient is C(4, 3)(-6)³ = 4 × (-216) = -864.

   λ = -864/α = -864/-24 = 36

4. So. λ = 36.

The common slip: -36. Dropping the minus sign from -6 in only one of the two coefficients gives -36, not 36. Both α = -24 and -864 are negative.

The idea to keep. Sum of coefficients means put x = 1. When a question removes one term from that sum, subtract that term's coefficient and use the number to pin down n.

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