Question

Suppose $2 - p$, $p$, $2 - \alpha$, $\alpha$ are the coefficients of four consecutive terms in the expansion of $(1 + x)^n$. Then the value of $p^2 - \alpha^2 + 6\alpha + 2p$ equals

• A. 4
• B. 10
• C. 8
• D. 6

Answer 1

Answer: (B)

10

Answer 2

Solution: Let the coefficients $2 - p, p, 2 - \alpha, \alpha$ be consecutive binomial coefficients:

$C_r = 2 - p, \quad C_{r+1} = p, \quad C_{r+2} = 2 - \alpha, \quad C_{r+3} = \alpha.$

Using the relationship for consecutive binomial coefficients:

- For $C_{r+1} = \frac{n - r}{r + 1} C_r$:

$p = \frac{n - r}{r + 1} (2 - p).$

Repeat for $C_{r+2}$ and $C_{r+3}$ to find $p$ and $\alpha$.

Substitute the values to find:

$p^2 - \alpha^2 + 6\alpha + 2p = 10.$