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

The Correct Option is B

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:

\[ p^2 - \alpha^2 + 6\alpha + 2p = 10. \]