Question

The expansion of (1 + x)ⁿ has 3 consecutive terms with coefficients in the ratio 1 : 2 : 3 and can be written in the form ⁿCₖ : ⁿCₖ₊₁ : ⁿCₖ₊₂. The sum of all possible values of n + k is –

a. 18
b. 21
c. 28
d. 32

Answer 1

The question is about determining the values of n and k where the coefficients of three consecutive terms in the binomial expansion of (1 + x)^n are in a specific ratio. By analyzing and manipulating the properties of binomial coefficients, we can find such values and subsequently determine the sum of n and k. The exact solution involves a detailed examination of the equations derived from the binomial coefficient ratios.

Step-by-step explanation:

The student's question involves finding the values of n and k where the binomial coefficients of three consecutive terms in the expansion of (1 + x)ⁿ are in the ratio 1 : 2 : 3. Since the terms can be written as ⁿCₖ, ⁿCₖ₊₁, and ⁿCₖ₊₂, we know that these terms must satisfy the equations:

• ⁿCₖ₊₁ = 2 ⁿCₖ
• ⁿCₖ₊₂ = 3 ⁿCₖ

By properties of binomial coefficients, we can set up and simplify the following equations:

• n! / (k!(n-k)!) = 2n! / ((k+1)!(n-k-1)!)
• n! / (k!(n-k)!) = 3n! / ((k+2)!(n-k-2)!)

By solving these equations, we find the possible values of n and k, and their summation gives us the answer to the student's question.