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

Given four consecutive binomial coefficients of \( (1 + x)^n \) in terms of variables \( p \) and \( \alpha \), we determine the value of a specific algebraic expression involving \( p \) and \( \alpha \).

Concepts Utilized:

1. Binomial Coefficients: \( C(n, r) \) or \( \binom{n}{r} \) represent the coefficients in the expansion of \( (1 + x)^n \), where \( r \) is the term index (starting from 0).

2. Pascal's Identity: The sum of two adjacent binomial coefficients equals the coefficient in the next row: \( \binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} \).

3. Symmetry of Binomial Coefficients: \( \binom{n}{r_1} = \binom{n}{r_2} \) implies \( r_1 = r_2 \) or \( r_1 + r_2 = n \).

Solution Steps:

Step 1: Express the four consecutive coefficients using binomial notation.

Let the coefficients correspond to terms \( k+1, k+2, k+3, \) and \( k+4 \). Their values are:

\[ \binom{n}{k} = 2 - p \] \[ \binom{n}{k+1} = p \] \[ \binom{n}{k+2} = 2 - \alpha \] \[ \binom{n}{k+3} = \alpha \]

Step 2: Apply Pascal's identity to establish relationships.

Summing the first two coefficients:

\[ \binom{n}{k} + \binom{n}{k+1} = (2 - p) + p = 2 \]

By Pascal's identity, \( \binom{n+1}{k+1} = 2 \).

Summing the last two coefficients:

\[ \binom{n}{k+2} + \binom{n}{k+3} = (2 - \alpha) + \alpha = 2 \]

By Pascal's identity, \( \binom{n+1}{k+3} = 2 \).

Step 3: Utilize the symmetry property to find a relation between \( n \) and \( k \).

From Step 2, \( \binom{n+1}{k+1} = \binom{n+1}{k+3} \). Since \( k+1 eq k+3 \), symmetry dictates:

\[ (k+1) + (k+3) = n+1 \] \[ 2k + 4 = n + 1 \implies n = 2k + 3 \]

Step 4: Determine the relationship between \( p \) and \( \alpha \).

Using \( n = 2k + 3 \) and the symmetry property \( \binom{n}{r} = \binom{n}{n-r} \):

\[ \binom{n}{k+1} = \binom{2k+3}{k+1} \] \[ \binom{n}{k+2} = \binom{2k+3}{k+2} = \binom{2k+3}{(2k+3) - (k+2)} = \binom{2k+3}{k+1} \]

Thus, \( \binom{n}{k+1} = \binom{n}{k+2} \). Substituting the given values:

\[ p = 2 - \alpha \implies p + \alpha = 2 \]

Step 5: Evaluate the target expression using the derived relationship.

The expression is \( p^2 - \alpha^2 + 6\alpha + 2p \). Substitute \( \alpha = 2 - p \):

\[ p^2 - (2 - p)^2 + 6(2 - p) + 2p \]

Expand and simplify:

\[ = p^2 - (4 - 4p + p^2) + (12 - 6p) + 2p \] \[ = p^2 - 4 + 4p - p^2 + 12 - 6p + 2p \]

Combine like terms:

\[ = (p^2 - p^2) + (4p - 6p + 2p) + (-4 + 12) \] \[ = 0 + 0 + 8 = 8 \]

Final Calculation & Result:

The relationship \( p + \alpha = 2 \) simplifies the expression.

Alternatively, rewrite the expression:

\[ (p^2 - \alpha^2) + 6\alpha + 2p = (p - \alpha)(p + \alpha) + 6\alpha + 2p \]

Substitute \( p + \alpha = 2 \):

\[ = (p - \alpha)(2) + 6\alpha + 2p = 2p - 2\alpha + 6\alpha + 2p = 4p + 4\alpha = 4(p + \alpha) \]

Substitute \( p + \alpha = 2 \):

\[ = 4(2) = 8 \]

The value of the expression is 8.